DENSITY DEPENDENT UTILITIES WITH ... - Math@LSU

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Communications on Stochastic Analysis Vol. 5, No. 4 (2011) 617-631

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DENSITY DEPENDENT UTILITIES WITH TRANSACTION COSTS ERIYOTI CHIKODZA AND JULIUS N. ESUNGE Abstract. We investigate the combined singular and impulse control problem in the context of jump diffusions. Problems of this nature often arise when both fixed and proportional transaction costs are considered, for instance, in finance. We formulate and prove a verification theorem for the generalized combined singular and impulse control. This theorem establishes sufficient conditions for the existence of both the value function and optimal combined controls. An illustrative example of this result is presented.

1. Introduction Several researchers have considered the problem of portfolio optimization in the presence of transaction costs, for example [1, 6, 7]. The inclusion of both fixed and proportional transaction costs gives rise to problems which exhibit both singular and impulse control features. Previous authors focused their work on such problems without considering the situation with jump diffusions. We seek to close this gap. In particular, using some of the earlier arguments, we develop the theory of combined singular and impulse control for Lévy processes. This paper is further distinguished by the fact that it illustrates the application of combined singular impulse control to the problem of optimal harvesting with density dependent prices, in a framework of jump diffusions and in the presence of transaction costs, with an example motivated by Example 3.1 in [2]. In [2] the problem of optimal stochastic harvesting with density dependent prices for diffusions is discussed under the no transaction costs assumption. For an extensive coverage of the theory and application of singular control and impulse control as separate stochastic control techniques for Lévy processes, we refer the reader to [12] and the references provided therein. The paper is organized as follows: we forumlate the general combined singular and impulse control problem in Section 2, followed by a discussion of the verification theorem in Section 3. The final section includes an example on the application of the theory of combined singular and impulse control for jump diffusions. This example takes both proportional and transaction costs into account. The present article is a substantial revision of [5]. Received 2011-4-4; Communicated by the editors. 2000 Mathematics Subject Classification. 49L20, 45K55, 60H10. Key words and phrases. Combined singular and impulse control, jump diffusion, variational inequalities. 617

618

ERIYOTI CHIKODZA AND JULIUS N. ESUNGE

2. Background and Problem Formulation Let (Ω, F , {Ft }t≥0 , P ) be a filtered complete probability space satisfying the usual conditions. Assume that in the absence of interventions, the state Y (t) ∈ Rk , of a given system evolves according to the following equations Z ˜ (dt, dz), γ(Y (t− ), z)N (2.1) dY (t) = b(Y (t))dt + σ(Y (t))dB(t) + Rk

k

k

k

Y (0− ) = y ∈ Rk , k×m

(2.2) k

d

k×d

where b : R → R , σ : R → R and γ : R × R → R are functions satisfying the conditions for the existence and uniqueness of a strong solution Y (t). ˜r (., .) Here, B(t) is m-dimensional Brownian motion with respect to {Ft } and N ˜ is a compensated Poisson random measure given by Nr (dt, dz) = Nr (dt, dz) − dtνr (dz); r = 1, 2, . . ., d where νr (.) is a Lévy measure associated with the Poisson random measure Nr (., .). A more extensive treatment of random measures and stochastic differential equations with a jump component may be found in [4, 8, 9, 11, 12, 13]. The generator L of Y (t) is given by Lφ(y) =

k k ∂φ ∂2φ X 1X bi y, u(y) (σσ T )is y, u(y) + ∂yi 2 i,s=1 ∂yi ∂ys i=1

+

Z

d n X φ(y) + γ (r) (y, u(y), z) − φ(y)

Rk r=1

o −∇φ(y)T γ (r) (y, u(y), z) νr (dzr ),

where φ ∈ C 2 (Rk ). Suppose that at any given point τj the decision maker is free to give the system an impulse, ξj ∈ Z ⊂ Rp , where Z is the set of all admissible impulses. The notion of an impulse control is defined below. Definition 2.1. An impulse control, for the system described by (2.1)-(2.2), is a double sequence v = (τ1 , τ2 , . . ., τj . . ., ξ1 , ξ2 , . . ., ξj , . . .)j≤M , M ≤ ∞, where 0 ≤ τ1 ≤ τ2 ≤ . . . is an increasing sequence of Ft -stopping times and ξ1 , ξ2 , . . . are the corresponding Fτj -adapted impulses at these stopping times. Let S ⊂ Rk be a fixed Borel set in which we seek solutions to the problem with S ⊂ S¯0 . Here S 0 denotes the interior of S and S¯0 is the closure of S 0 . Given continuous functions f : S → R, g : Rk → R, κ = [κie ] ∈ Rk×p and θ = [θi ], let the profit of making an intervention with impulse ξ ∈ Z when the state is y be K(y, ξ), where K : S × Z → R. Definition 2.2. Let H be the space of all measurable functions h : S → R. The intervention operator M : H → H is defined by Mh(y) = sup{h(Γ(y, ξ)) + K(y, ξ); ξ ∈ Z}.

DENSITY DEPENDENT UTILITIES WITH TRANSACTION COSTS

619

Suppose that at times tn ∈ [τj , τj+1 ] one is allowed to apply the singular control ψ for n = 1, 2, . . ., q, where ψ ∈ Rp is an adapted c´ adl´ ag process with nonnegative, increasing components such that ψ(0− ) = 0. Let the jumps caused by the singular control ψ be denoted by △ψ Y (t) = κ(Y (t− ))△ψ

and consider

△ψ φ(Y (tn )) = φ(Y (tn )) − φ(Y (t− n )) to be the change in φ resulting from the jump △ψ(t) = ψ(t) − ψ(t− ) at t = tn . Denote by W the set of all admissible combined controls w = (v, ψ). Suppose that the controlled process Y (w) satisfies (2.4)-(2.5) given as follows Y (w) (0− ) = y and Y (w) (t) = Y (t), Y

(w)

dY

(τj ) = Γ(Yˇ (w) (τj− ), ξj ),

(w)

(t) = b(Y Z +

(w)

(t))dt + σ(Y

0 < t < τ1 ,

(2.3)

j = 1, 2, . . . (w)

(2.4) w

(t))dB(t) + κ(Y (t))dψ

˜ (dt, dz), γ(Y (w) (t− ), z)N

τj < t < τj+1 < τ ∗ ,

(2.5)

Rl

where

τ ∗ = τ ∗ (ω) = lim (inf{t > 0; | Y (v) (t) |≥ R}) ≤ ∞. R→∞

In equations (2.4)-(2.5) above, Yˇ (w) (τj− ) = Y (w) (τj− ) + △N Y (τj ) defines the jump in Y (w) (τj ) which stems from N (., .) and Γ : Rk × Z → Rk is a given function. Let τS := inf{t ≥ 0; Y (w) ∈ / S} and consider T := {τ ; τ stopping times, 0 ≤ τ ≤ τS }.

Define a performance functional J (w) , for the controlled process Y (w) , by hZ τS (w) y f (Y (w) (t))dt + g(Y (w) (τS ))χ{τS 0 before reaching zero. This in turn gives analytical insight as to why the Bessel processes of order 5, 3, and −1 together with a squared one-dimensional Brownian motion are intrinsically related. As a final example we study the non-homogeneous Ornstein-Uhlenbeck process.

1. Introduction Alternative representations of a problem are often used in Stochastic Analysis. Such is the case of Girsanov’s theorem—removal of a drift [see Section 3.5 in Karatzas and Shreve [10]]—Doob’s h-transform [see Chapter 6 in Rogers and Williams [15]], or the absolute continuity property for the laws of the Bessel processes with different indices [see Chapter XI in Revuz and Yor [14]]. In this work we derive three systems of coupled Kolmogorov backward and forward PDEs which are used to derive changes of measure and/or dual representations of SDEs. [see Hernandez-del-Valle [6] for an application of similar techniques in hitting densities of Brownian motion, Hernandez-del-Valle [7] for alternative results on the procedure described within, or Hernandez-del-Valle [8] for an application in the 3-D Brownian motion case]. Applications of the techniques described in this work is finding the transition probabilities of, at first sight, “strange” SDEs [see Section 5]. For instance, we find the density of a one-dimensional Wiener Received 2011-6-1; Communicated by the editors. 2000 Mathematics Subject Classification. Primary: 37A50, 60G07, 60H30. Key words and phrases. Doob’s h-transform, change of measure, Bessel process, first hitting time. * The research of the author was partially supported by Algorithmic Trading Management LLC. 701

702

GERARDO HERNANDEZ-DEL-VALLE

process started within a range [0, a] and that hits level a for the first time at time v > 0 (before ever reaching level 0). As an application of one of the coupled PDE systems, we obtain (for a special case) Bessel’s absolute continuity property, as well as a PDE interpretation of the so-called “Bessel process with negative dimension” [see Chapter XI, Section 1 in Revuz and Yor [14], and G¨ oing-Jaeschke and Yor [5]]. We also illustrate the results by deriving the density of the first time that a 5-D Brownian motion hits a ball from within: (1) in terms of a functional of a one-dimensional Brownian motion absorbed at 0 and a, and (2) in terms of the mean of a cubed Bessel process with dimension d = −1, absorbed at 0 or a. The paper is organized as follows. In Sections 2 and 3, equivalent systems of backward and forward PDEs are derived. These in turn are used to derive changes of measure and/or dual representations of SDEs. Next, in Sections 4 and 5, the results in Section 2 are used to derive: (a) the transition density of a Brownian motion, started at y (for 0 ≤ y ≤ a), which is absorbed at 0 and that reaches level a at a time u, and (b) an alternative derivation of the density of a 5-D Wiener process hitting a ball from within. In Section 6, we relate Bessel process of order 5 and −1 and find the density of the first time that a Bessel −1 hits level a > 0 before reaching zero. To this end, we make use of the ideas developed Section in 5. We conclude in Section 7 with some final comments. 2. Coupled Systems of PDEs I In our results below we consider real valued differentiable functions h, u, v, and w, depending on variables (t, x; s, y) ∈ R+ × R × R+ × R. As in the analysis of diffusion processes, PDEs with derivatives with respect to (t, x) are called backward equations, whereas PDEs with derivatives in (s, y) are called forward equations. In this section we relate systems of coupled backward and forward Kolmogorov equations, through a pair of backward equations [see (2.5.a) and (2.7.a), or equivalently (2.9.a) and (2.11.a)]. This in turn allows us to transform problems of diffusions with drift (or convective mass transfer) into problems of diffusions without drift (or with potential), under a suitable change of measure. In particular, the drift is at most a sum of two functions which in turn are solutions to backward heat equations. We will clarify the last sentence of the previous paragraph with the following example. Example 2.1. (i) (Bessel process of order 3.) First note that h(t, x) = x is a solution to the backward heat equation −ht =

1 hxx . 2

Let us suppose that the process X satisfies that dXt =

1 dt + dWt , Xt

X0 = x > 0.

(2.1)

CHANGES OF MEASURE AND BESSEL PROCESSES

703

Since hx /h = 1/x we may express (2.1) as dXt =

hx (t, Xt ) dt + dWt , h(t, Xt )

X0 = x > 0.

(ii) (Bessel process of order 5.) Suppose the process Y is such that dYt =

2 dt + dWt , Yt

Y0 = y > 0,

(2.2)

and set k(t, x) = h(t, x) = x. Then we may express (2.2) as hy (t, Yt ) ky (t, Yt ) dYt = + dt + dWt , Y0 = y > 0. h(t, Yt ) k(t, Yt ) (iii) (Brownian bridge.) Suppose Z has the dynamics dZt = − and set

Zt dt + dWt , s−t

t ≤ s,

(2.3)

1 x2 q(t, x) = p exp − . 2(s − t) 2π(s − t)

Since qx /q = −x/(s − t), it follows that we may express (2.3) as dZt =

qz (t, Zt ) dt + dWt , q(t, Zt )

t ≤ s.

Proposition 2.2. Let σ : R+ → R, and h, v, w and k be of class C 1,2 . Suppose that v, w, and h satisfy the following identity v(t, x) =

w(t, x) h(t, x)

(2.4)

and h(t, x) 6= 0 for some strip in R+ × R. If h and v satisfy respectively (a) and (b)  −ht = 12 σ 2 hxx (a)    h k 1 2 −vt = 2 σ vxx + σ 2 hx + kx vx (2.5) (b)    −wt + σ 2 kkx hhx w = 12 σ 2 wxx + σ 2 kkx wx , (c) then (c) holds.

Proof. From (2.4) and (2.5.b) we have ht w − hwt h2

=

1 2 wxx h − whxx hx (wx h − whx ) σ − σ2 3 2 h2 h hwx − hx w 2 hx (wx h − whx ) 2 kx +σ +σ . h3 k h2

Equivalently ht w − hwt h2

=

1 2 wxx h − whxx kx σ + σ2 2 h2 k

hwx − hx w h2

.

704


In turn, from (2.5.a) it follows that −

wt 1 wxx kx wx kx hx = σ2 + σ2 − σ2 w. h 2 h k h k h2

Proposition 2.3. Let σ : R+ → R, and u, w and k be of class C 1,2 . Suppose that w, u, and k satisfy the following identity w(t, x) =

u(t, x) k(t, x)

(2.6)

and k(t, x) 6= 0 for some strip in R+ × R. If k and w satisfy respectively (a) and (b)  (a) −kt = 12 σ 2 kxx    −ut + σ 2 kkx hhx u = 12 σ 2 uxx (b) (2.7)    1 2 2 kx hx 2 kx −wt + σ k h w = 2 σ wxx + σ k wx (c) then (c) holds.

From Propositions 2.2 and 2.3 we have the next corollary. Corollary 2.4. Let h, v, k and u be of class C 1,2 as well as solutions to (2.5.a), (2.5.b), (2.7.a), and (2.7.b), respectively. Then they satisfy the identity v(t, x) =

u(t, x) k(t, x) · h(t, x)

given that k(t, x) · h(t, x) 6= 0 for some strip in R+ × R. Remark 2.5. Propositions 2.2 and 2.3 suggest that for j = 1, . . . , n and given δj (t, x) =

hjx (t, x) hj (t, x)

where each hj is a solution to −hjt =

1 j h . 2 xx

A backward equation of the following type   n X 1 −vt = vxx +  δj (t, x) vx 2 j=1

may be solved using the procedure described above.

Proposition 2.6. Let h, v and w be of class C 1,2 . Suppose that v, h, and w satisfy the following identity v(s, y) = h(s, y) · w(s, y)

(2.8)


and h(t, x) · k(t, x) 6= 0 for some strip in R+ × R. If h (a) and (b)  −hs = 21 hyy    h i  hy ky ∂ vs = 12 vyy − ∂y + v h k  h i h i    w + ky hy w = 1 w − ∂ ky w , s yy k h 2 ∂y k

705

and v satisfy respectively (a) (b)

(2.9)

(c)

then (c) holds.

Proof. From (2.8) and (2.9.b) we have hs w + hws

=

=

=

=

1 1 hyy w + hy wy + hwyy 2 2 ∂ hy ky hw − + ∂y h k 1 1 hyy w + hy wy + hwyy 2 2 ∂ ∂ ky − [hy w] − hw ∂y ∂y k 1 1 hyy w + hy wy + hwyy 2 2 ky ∂ ky w h − hy w −hyy w − hy wy − ∂y k k 1 1 ∂ ky ky − hyy w + hwyy − w h − hy w. 2 2 ∂y k k

Equivalently, from (2.9.a), hws +

ky hy w k

=

1 ∂ ky hwyy − w h. 2 ∂y k

Proposition 2.7. Let k, u and w are of class C 1,2 . Suppose that w, k, and u satisfy the following identity w(s, y) = k(s, y) · u(s, y) +

and k(t, x) · h(t, x) 6= 0 for some strip in R × R. If k (a) and (b)  −ks = 21 kyy     h i k h us + ky hy u = 12 uyy  i h i h    ws + ky hy w = 1 wyy − ∂ ky w , k h 2 ∂y k

(2.10) and u satisfy respectively (a) (b) (c)

then (c) holds.

From Propositions 2.6 and 2.7 we have the next corollary.

(2.11)

706


Corollary 2.8. Let h, v, k and u be of class C 1,2 , as well as solutions to (2.9.a), (2.9.b), (2.11.a), and (2.11.b), respectively. Then they satisfy the identity v(s, y) = h(s, y) · k(s, y) · u(s, y). Theorem 2.9. Let h and k be of class C 1,2 (R+ × R) as well as solutions to the backward equations (2.5.a) and (2.9.a) [or equivalently (2.7.a) and (2.11.a)] respectively. Furthermore, consider processes X, Y , and Z, which respectively satisfy (at least in the weak sense), the following equations (each under their corresponding ˜ and Q), measures P, Q, hx (t, Xt ) kx (t, Xt ) (P) dXt = + dt + dBt h(t, Xt ) k(t, Xt ) kx (t, Zt ) ˜ (Q) dZt = dt + dBt (2.12) k(t, Zt ) (Q)

dYt

=

dBt .

Moreover, suppose that f (τ, z) =

kz (τ, z) hz (τ, z) k(τ, z) h(τ, z)

(2.13)

and k(τ, z) · h(τ, z) 6= 0 for some strip in R+ × R. Then the following identities hold Pt,x (Xs ∈ A) Z s ˜ h(s, Zs ) = EQ exp − f (τ, Z )dτ I τ (Zs ∈A) z h(t, z) t Z s k(s, Ys ) h(s, Ys ) = EQ exp − f (τ, Y )dτ I τ (Ys ∈A) . y k(t, y) h(t, y) t

(2.14)

Proof. Let X be a process with dynamics as in (2.12.P). Hence its transition density G is simultaneously a solution of (2.5.b), in the backward variables (t, x), and a solution of (2.9.b), in the forward variables (s, y). In turn, from (2.4) and (2.8) we have w(t, x) , h(t, x) G(t, x; s, y) =: v(s, y) = h(s, y) · w(s, y), G(t, x; s, y) =: v(t, x) =

respectively. Next, if we let H be simultaneously H(t, x; s, y) =: w(t, x)

and

H(t, x; s, y) =: w(s, y),

then we have G(t, x; s, y) =

h(s, y) H(t, x; s, y). h(t, x)


707

That is, if G is the transition density of process Z, which has dynamics as in ˜ it follows (2.12.Q) Z Pt,x (Xs ∈ A) = G(t, x; s, y)dy A Z h(s, y) H(t, x; s, y)dy = A h(t, x) Z h(s, y) H(t, x; s, y) = G(t, x; s, y)dy A h(t, x) G(t, x; s, y) h(s, Zs ) ˜ = EQ Z I s (Zs ∈A) . t,x h(s, x) ˜ To this end, and given that f is as in Our goal is to characterize Z under Q. (2.13), we have from Itˆ o’s lemma Z s d h(s, Zs ) exp − f (τ, Zτ )dτ t Z s 1 kz = exp − f (τ, Zτ )dτ hs − f h + hzz + hz ds + hz dBs 2 k Zt s = exp − f (τ, Zτ )dτ t 1 kz kz hz × hs − h + hzz + hz ds + hz dBs kh 2 k Z s = exp − f (τ, Zτ )dτ hz dBs . t

hence h(·, Z· ) exp{− ˜

EQ t,x This yields

˜ f (τ, Zτ )dτ } is a Q-martingale, and such that Z s h(s, Zs ) exp − f (τ, Zτ )dτ = 1. h(t, z) t

R· t

Z Zs = exp −

t

s

f (τ, Zτ )dτ

.

The second identity follows using similar arguments.

3. Coupled Systems of PDEs II In this section we relate systems of coupled backward and forward Kolmogorov equations, through a backward equation [see (3.5.a), or equivalently (3.7.a)]. This in turn allows us to obtain a dual representation of SDE problems under a proper change of measure. We first give an example to illustrate this idea. Example 3.1. Let process X and Y be such that 2 dXt = dt + dBt Xt 1 dYt = − dt + dBt . Yt

708


It is known that these two processes are related; X is a Bessel process of order 5, and Y of order −1; see Göing-Jaeschke and Yor [5]. We will give a PDE interpretation of how these two processes are linked. To do so, let us consider the backward Kolmogorov equations associated to the processes X and Y , namely, 1 2 vxx + vx 2 x 1 1 (Y ) − wt = wxx − wx . 2 x Next note that the sum of the drifts of X and Y , which add up to (X)

− vt

=

(3.1) (3.2)

1 3 2 + = , x x x can be expressed in terms of functions h and p, which are solutions of −ht =

1 1 hxx − hx 2 x

and

− pt =

1 pxx , 2

(3.3)

respectively. To do so, let p = x and h = x3 , i.e., 2 px hx + = . x p h In particular, in this section we will show, that for real valued functions v, w, and h which satisfy respectively (3.1), (3.2), and (3.3) the following identity holds h·v

= w.

In turn, the previous relationship will allow us to give a systematic procedure to transform a process X (with drift) into another process Y (with drift) under a suitable change of measure. Proposition 3.2. Let α : R → R be a real valued function, and h, v and w be of class C 1,2 . Suppose that v, w, and h satisfy the following identity v(t, x) =

w(t, x) h(t, x)

and h(t, x) 6= 0 for some strip in R+ × R. If h and (b)  −ht = 12 hxx + α(x)hx    −vt = 12 vxx + α(x) + hhx vx    −wt = 21 wxx + α(x)wx ,

(3.4) v satisfy respectively (a) and (a) (b) (c)

then (c) holds.

Proof. From (3.4) and (3.5.b) we have ht w − hwt h2

=

1 wxx h − whxx hx (wx h − whx ) − 2 h2 h3 hx (wx h − whx ) wx h − whx + + α(x) , 3 h h2

(3.5)


709

alternatively 1 wxx h − 2 Rearranging terms it follows that 1 −h wt + wxx + α(x)wx 2 ht w − hwt

=

1 whxx + α(x)wx h − α(x)whx . 2

1 = −w ht + hxx + α(x)hx . 2

Proposition 3.3. Let α : R → R be a real valued function, and h, v and w be of class C 1,2 . Suppose that v, w, and h satisfy the following identity v(s, y) = w(s, y) · h(s, y).

and h(t, x) 6= 0 for some strip in R+ × R. If h and v (b)  −hs = 12 hyy + α(y)hy    h i h ∂ vs = 12 vyy − ∂y α(y) + hy v    ∂ [α(y)w] ws = 21 wyy − ∂y

(3.6) satisfy respectively (a) and (a) (b)

(3.7)

(c)

then (c) holds.

Proof. From (3.6) and (3.7.b) we have 1 1 ws h + whs = wyy h + wy hy + whyy 2 2 i ∂ ∂ h − [α(y)wh] − hy w ∂y ∂y 1 1 = wyy h + wy hy + whyy 2 2 ∂ − [α(y)w] h − wα(y)hy − hyy w − hy wy ∂y 1 1 ∂ = wyy h − whyy − [α(y)w] h − wα(y)hy . 2 2 ∂y Equivalently, rearranging terms we have ∂ 1 1 h ws − wyy + [α(y)w] = −w hs + hyy + α(y)hy . 2 ∂y 2

Theorem 3.4. For a given α : R → R let h be a solution to (3.5.a) [or equivalently (3.7.a)]. Furthermore, let processes X and Y be well defined (at least in the weak sense), with the following dynamics (each under their corresponding measures P, and Q), hx (t, Xt ) (P) dXt = α(Xt ) + dt + dBt h(t, Xt ) (Q) dYt = α(Yt )dt + dBt .

710


Then the following identity holds Pt,x (Xs ∈ A) =

EQ y

h(s, Ys ) I(Ys ∈A) . h(t, y)

Proof. The proof is carried out in the same way as the proof of Theorem 2.9. That is, It follows from Kolmogorv’s backward and forward equations together with Propositions 3.2 and 3.3. Corollary 3.5. Under the assumptions of Theorem 3.4, the process h(·, Y· ) is a Q martingale. Proof. This corollary follows from Itô’s lemma, the fact that h is a solution to (3.5.a), and the equality dYt = α(Yt )dt + dBt . 4. Bessel Process I In this section, making use of the ideas presented in Section 3, we revisit the problem of finding the density of the first time that a 5-D Brownian motion hits a ball from within. Our derivation, however, is new and does not make use of a Laplace-Gegenbauer transform as in, Wendel [17], Hsu [9], Yin [18], and Betz & Gzyl [1, 2]. Recent results on the subject can also be found in Byczkowki et. al [4] and references therein. Standard references in the theory of Bessel processes are Knight [11], Ray [13], Lamperti [12] or Revuz and Yor [14]. Remark 4.1. In the next two sections we will make use of the following notation: 1 (y − z)2 p(t; y, z) := √ exp − , (4.1) 2t 2πt p− (t; y, z) := p(t; y, z) − p(t; y, −z). In addition, if T and T0 are respectively the first time that a one-dimensional standard Brownian motion B started at y reaches level a and level 0 respectively, let " ∞ X 1 Pt,y (T ∈ s, T0 > s) := p (2na + a − y) 2π(s − t)3 n=−∞ # (2na + a − y)2 × exp − , (4.2) 2(s − t) and Pt,y (Bs ∈ dx, T ∧ T0 > s) :=

∞ X

n=−∞

p− (s − t; y, x + 2na)dx,

(4.3)

which is the probability of a Brownian motion B started at y ∈ (0, a) before reaching levels 0 or a. Remark 4.2. Let X be a Bessel process such that 2 dXt = dt + dBt . Xt


711

Next, note that h(t, x) = k(t, x) = x are solutions to the backward heat equation (2.5.a) and (2.7.a) respectively (with σ 2 = 1). Hence the latter SDE for X can be expressed, in terms of h and k, as hx (t, Xt ) kx (t, Xt ) dXt = + dt + dBt . (4.4) h(t, Xt ) k(t, Xt ) From Theorem 2.9 it follows that process X is a Q-Wiener process dYt = dBt

(4.5)

absorbed at 0. For X as in (4.4) and X0 < a, let T := inf {t ≥ 0|Xt = a} .

(4.6)

From (2.14), in Theorem 2.9, the probability of hitting a, starting from within the ball, satisfies the identity 2 Z s 1 Q Ys P(T < t) = Ey du I(T0 >s,T s,T u) . 2 y ˜2 0 y 0 Yu

EQ y

where Y˜ is a process which starts at y, is absorbed at zero, and reaches a, for the first time at t = u.

712


Proof. Let M = Y 2 · R. Then 2 Z s 1 Q Ys Ey − du I(T0 >s) I(T s) I(T s) I(T s) I(T s) I(T s,T v exp T 0 2 y 2 0 y 0 Yu ×Py (T ∈ dv, T0 > v) Z v Z t 2 a Q 1 du Py (T ∈ dv, T0 > v) . = E − exp 2 y ˜2 0 y 0 Y u

5. Dynamics of the Process Y˜ In this section we derive the dynamics of process Y˜ . Proposition 5.1. Let Pt,x (T ∈ s, T0 > s)

(5.1)

be the probability density of a one-dimensional standard Brownian motion absorbed at the origin defined in (4.2). Since (5.1) is a function of only s − t and x let ha (s − t, x) := Pt,x (T ∈ s, T0 > s).

Then Y˜ , which is a process absorbed at 0 and that reaches a for the first time at t = s (and given that hy stands for differentiation with respect to y), satisfies that dY˜t

=

Y˜s

=

hay (s − t, Y˜t ) dt + dWt , ha (s − t, Y˜t )

a.

0 < t < s,

(5.2)


713

In particular it has the following transition density G(t, y; τ, x) =

ha (s − τ, x) Pt,y (Yτ ∈ dx, T ∧ T0 ) , ha (s − t, y)

(5.3)

where Pt,y (Yτ ∈ dx, T ∧ T0 ) is defined in (4.3).

Proof. We will construct the finite dimensional distributions of an absorbed Brownian motion at 0, and that reaches level a for the first time at t = s. Given t0 < t1 < · · · < t = s, we shall compute Py (Yt1 ∈ dy1 , Yt2 ∈ dy2 , . . . , Ytn ∈ dyn |T = s, T0 > s) Py (Yt1 ∈ dy1 , Yt2 ∈ dy2 , . . . , Ytn ∈ dyn , T ∈ ds, T0 > s) = . Py (T ∈ ds, T0 > s)

(5.4)

By the independence of increments of Brownian motion Y , the numerator of the previous expression is Pt0 ,y (Yt1 ∈ dy1 , T ∧ T0 > t1 )Pt1 ,y1 (Yt2 ∈ dy2 , T ∧ T0 > t2 ) × · · · · · · × Ptn−1 ,yn−1 (Ytn ∈ dyn , T ∧ T0 > tn )Ptn ,yn (T ∈ ds, T0 > s). This in turn implies that (5.4) equals n Ptn ,yn (T ∈ ds, T0 > s) Y Pt ,y (Yt ∈ dyj , T ∧ T0 > tj ). Py (T ∈ ds, T0 > s) j=1 j−1 j−1 j

(5.5)

However, since

Ptn ,yn (T ∈ ds, T0 > s) Py (T ∈ ds, T0 > s) Pt ,y (T ∈ ds, T0 > s) Ptn−1 ,yn−1 (T ∈ dtn , T0 > tn ) · ··· = n n Py (T ∈ ds, T0 > s) Ptn−1 ,yn−1 (T ∈ dtn , T0 > tn ) Pt ,y (T ∈ ds, T0 > s) ··· 1 1 Pt1 ,y1 (T ∈ ds, T0 > s) n Y Ptj ,yj (T ∈ s, T0 > s) = , P t j−1 ,yj−1 (T ∈ s, T0 > s) j=1 (5.5) equals n Y

Ptj ,yj (T ∈ s, T0 > s) Pt ,y (Yt ∈ dyj , T ∧ T0 > tj ). P (T ∈ s, T0 > s) j−1 j−1 j j=1 tj−1 ,yj−1 On the other hand, to show that equation (5.3) is the transition density of Y˜ , with dynamics as in (5.2), we make use of Propositions 2.3 and 2.7 by noting that ha is a solution to the backward equation (2.7.a) [or equivalently (2.11.a)] (with σ = 1 and h = 0). Proposition 5.2. Let Y˜ be a process with dynamics as in (5.2). Then Z s 1 u(t, y) = EQ exp − du t,y ˜2 t Y u

(5.6)

714


is equivalent to the following Cauchy problem 1 u = y2 u(s, y) =

−ut +

hay (s − t, y) 1 uyy + a uy , 2 h (s − t, y) 1.

(5.7)

Furthermore, u=

v , ha

where −vt +

1 v y2

=

1 vyy 2

(5.8)

Proof. The equivalence between (5.6) and (5.7) follows from the Feynman-Kac theorem [see for instance pp. 366–367, Theorem 5.7.6 in Karatzas and Shreve [10]]. On the other hand equation (5.8) follows from Proposition 2.3 and by noting that ha is a solution of the backward equation (2.7.a) [or equivalently (2.11.a)] (with σ = 1 and h = 0). Theorem 5.3. Let X be as in (4.4) and T as in (4.6). Then the probability that a 5-D Brownian motion started at 0 hits a ball of radius a is P(T ∈ dt)/dt ( ) ∞ X 1 λ2j 2 λ3j exp − 2 u = 3 a2 2a j=1 " # (1 − λ2j ) sin(λj ) − λj cos(λj ) × , 2λ2j + λj sin(2λj ) + 2 cos(2λj ) − 2

(5.9)

where λj is the j-th root of the Bessel function J3/2 of the first kind. Proof. Given a variable τ which satisfies 0 ≤ t ≤ τ ≤ s, let δ = τ − t. Then equation (5.8) becomes 1 1 vδ = vyy − 2 v. (5.10) 2 y For constants µ > 0 and A ∈ R, a solution to (5.10) is given by 1 2 √ v(δ, y) = Ae− 2 µ δ xJ3/2 (µy). In fact (in terms of t) the following linear combination is also a solution ∞ y λ2 X 1 j √ v(t, y) = cj e− 2 a2 (τ −t) yJ3/2 λj . a j=1

Let the previous function satisfy the following boundary condition at t = τ u(τ, y) = ha (s − τ, y).

From the two previous expressions we have ∞ y X √ ha (s − τ, y) = cj yJ3/2 λj . a j=1


715

Hence by the orthogonality properties of the Bessel function J3/2 , we have Z a Z a x x √ a 2 dx = cj x · J3/2 λj dx. x · h (s − τ, x)J3/2 λj a a 0 0 This implies that Ra√ a xh (s − τ, x)J3/2 λj xa dx 0 Ra cj = . x 2 0 x · J3/2 λj a dx

Since we will only study the case when y = 0, we will make use of the following fact in the sequel r 1 3/2 2 −3/2 . lim y J3/2 (µy) = µ y→0 3 π If we let bj = λj /a d √ pj (x) := xJ3/2 (bj x) dx r 2 1 2 2 = (bj x − 1) sin(bj x) + bj x cos(bj x) . 3/2 π b x2 j

Given that p is as in (4.1), we need to find the limit as τ → s of Z a x √ x · ha (s − τ, x)J3/2 λj dx a 0 ∞ Z a x X √ = x · J3/2 λj dp(s − τ ; 2na + a, x) a n=−∞ 0 ∞ Z a X =− pj (x)p(s − τ ; 2na + a, x)dx, n=−∞

0

The second line of the previous expression follows from the integration by parts √ formula and by noting that J3/2 (λj ) = 0, and on the other extreme we have 0 for all n. Next we make the following change of variable 2na + a − x z= √ . s−τ That is, for each pj Z √2na s−τ √ 1 z2 √ e− 2 pj (2na + a − z s − τ )dz. − 2na+a 2π √ s−τ However, notice that as τ → s the only term that survives is when n = 0 in which case we have Z 0 √ 1 z2 1 √ e− 2 pj (a − z s − τ )dz = pj (a). − 2na+a 2 2π √ s−τ Equivalently, 1 pj (a) 2

=

√

1 1 2 (λj − 1) sin(λj ) + λj cos(λj ) . 3/2 2πa λj

716


Since Z

a

0

2 x J3/2 (bj x) dx

2a2 b2j + abj sin(2abj ) + 2 cos(2abj ) − 2 2πab3j

=

= a2 · we conclude that cj =

" √ 3/2 2πλj a5/2

2λ2j + λj sin(2λj ) + 2 cos(2λj ) − 2 , 2πλ3j

# (1 − λ2j ) sin(λj ) − λj cos(λj ) , 2λ2j + λj sin(2λj ) + 2 cos(2λj ) − 2

which in turn yields ∞ X 2 λ3j j=1

(

1 λ2j exp − u 3 a4 2 a2

)"

(1 − λ2j ) sin(λj ) − λj cos(λj ) 2λ2j + λj sin(2λj ) + 2 cos(2λj ) − 2

#

or after multiplying the previous expression by a2 ( )" # ∞ X (1 − λ2j ) sin(λj ) − λj cos(λj ) 2 λ3j 1 λ2j exp − 2 u , 3 a2 2a 2λ2j + λj sin(2λj ) + 2 cos(2λj ) − 2 j=1 which is (5.9).

6. Bessel Process II

In this section we will make use of the ideas presented in Section 4 to find the probability that a Bessel process with negative order hits level a > 0 before hitting zero. Theorem 6.1. Suppose X and Y satisfy the equations: 2 (P) dXt = dt + dBt , Xt 1 (Q) dYt = − dt + dBt . Yt Let T be as in (4.6) and, under Q, T0 = inf{t ≥ 0|Yt = 0}.

Then Py (T < t) = EQ y In particular,

3 h(t, Yt ) Yt I(T t) = EQ I (T t) . y h(0, Y0 ) y3

Py (T < t) =

a3 Q (T < t, T0 > t) . y3

Proof. Let α(x) = −1/x. Then by equation (3.5) in Proposition 3.2, we have  −ht = 12 hxx − x1 hx    −vt = 12 vxx + − x1 + hhx vx    −wt = 12 wxx − x1 wx .


717

In particular, a solution to (3.5.a) is h(x) = x3 . Hence  −ht = 12 hxx − x1 hx    −vt = 12 vxx + x2 vx    −wt = 12 wxx − x1 wx . From (3.4) we know that

w w = 3 h x Alternatively, from Theorem 3.4, the processes X and Y that satisfy 2 (P) dXt = dt + dBt Xt 1 (Q) dYt = − dt + dBt Yt are related by 3 h(t, Yt ) Q Yt Py (T > t) = EQ I = E . I y y h(0, Y0 ) (T >t,T0 >t) y 3 (T >t,T0 >t) v=

From Itˆ o’s Lemma dYt3 = 3Yt2 dYt + 3Yt dt = −3Yt2 dBt .

Thus Y 3 is a martingale and hence, from the optional sampling theorem EQ Yt3 I(T t) = EQ EQ Yt3 I(T t) |Ft∧T0 ∧T = EQ YT30 ∧T ∧t I(T t) =

a3 Q (T < t, T0 > t) .

Example 6.2. In Salminen [16] the author derives the density of a process X with dynamics 1 dXt = bXt + dt + dBt . Xt This process X can be interpreted as the Euclidean norm of three independent copies of an Ornstein-Uhlenbeck process with dynamics dUt = bUt + dWt . Making use of Theorem 3.4 in Section 3 we arrive at the same conclusion. Namely, given processes X and Y with dynamics 1 (P) dXt = bXt + dt + dBt Xt (Q) dYt = bYt + dBt the following identity holds h i −b(s−t) Pt,x (Xs ∈ A) = EQ Y e I t (Ys ∈A,T0 >s) , y

718


where T0 := inf {t ≥ 0|Yt = 0} . To do so, in (3.5)  −ht = 12 hxx + α(x)hx    −vt = 12 vxx + α(x) + hhx vx    −wt = 12 wxx + α(x)wx

(6.1)

set α(x) = bx and as a solution of

−ht =

1 hxx + bxhx 2

choose h(t, x) = xe−bt . Hence hx /h = 1/x and thus −vt

hx 1 vxx + α(x) + vx = 2 h 1 1 = −vt = vxx + bx + vx . 2 x 7. Concluding Remarks

In this work we derive systems of coupled backward and forward Kolmogorov equations which in turn are used to derive dual representations of SDEs under a proper change of measure. In particular, the local drift of the corresponding process must be a linear combination of solutions of backward heat equations. Examples of process with this property are: Brownian bridge, 3-D Bessel bridge, Bessel processes of order 1 and 2. We also derive, making use of the techniques described within, the density of a Brownian motion started within some range [0, a], which hits level a at time v > 0 before ever reaching zero. We also derive, without making use of a Laplace-Gegenbauer transform, the density of the first time that a Bessel process of order 5 hits level a > 0. Our function is the same as the one appearing in equation (2.0.2) p. 398 in Borodin and Salminen [3]. Next, we relate the previous result, with the density of the first time that a Bessel of order −1 hits level a before reaching zero. Regarding a Bessel process of negative order see Göing-Jaeschke and Yor [5], and Revuz and Yor [14]. The coupled systems of one-dimensional backward and forward Kolmogorov equations, described in this paper, seem to have a “nice” and useful analytical structure. The extension of these results to d-dimensional PDEs as well as its implementation is work in progress. References 1. Betz, C. and Gzyl, H.: Hitting spheres from the exterior. Ann. Probab. 22 (1994) 177–179. 2. Betz, C. and Gzyl, H.: Hitting spheres with Brownian motion and Sommerfeld’s radiation condition. J. Math. Anal. Appl. 182 (1994) 301–308. 3. Borodin, A. N. and Salminen, P.: Handbook of Brownian Motion-Facts and Formulae, Birkh¨ auser Verlag, Basel-Boston-Berlin, second edition, 2002. 4. Byczkowski, T., Malecki, J. and Ryznar, M.: Hitting half-spaces by Bessel-Brownian diffusions. Potential Anal. 33 (2010) 47–83.


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5. G¨ oing-Jaeschke, A. and Yor, M.: A survey and some generalizations of Bessel processes. Bernoulli 9 No. 2 (2003) 313–349. 6. Hernandez-del-Valle, G.: On martingales, Bessel processes, and Durbin’s hitting time equation (2010) Submitted. 7. Hernandez-del-Valle, G.: On Doob’s h-transform, Girsanov’s, and transition densities (2011) Submitted. 8. Hernandez-del-Valle, G.: On the density of the first time that a 3-D Brownian motion hits a sphere (2011) Submitted. 9. Hsu, P.: Brownian exit distribution of a ball. Seminar on Stochastic Processes, (1986) 108– 116, Birk¨ auser, Boston,. 10. Karatzas, I. and Shreve, S.: Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. 11. Knight, F. B.: Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 107 (1963) 56–86. 12. Lamperti, J.: Semi-stable Markov processes. Z. Wahr. Verw. Geb. 22 (1972) 205–225. 13. Ray, D. B.: Sojourn times of diffusion processes. Illinois J. Math. 7 (1963) 615–630. 14. Revuz, D., and Yor, M.: Continuous Martingales and Brownian Motion, Springer-Verlag, New York, 2005. 15. Rogers, L. C. G. and Williams, D.: Diffusions, Markov Processes and Martingales: Volume 2, Ito Calculus, Cambridge University Press, Cambridge, UK, 2000. 16. Salminen, P.: On conditional Ornstein-Uhlenbeck processes, Adv. in Appl. Probab. 16 No. 4 (1984) 920–922. 17. Wendel, J. G.: Hitting spheres with Brownian motion, Ann. Probab. 8 (1980) 164–169. 18. Yin, C. C.: The joint distribution of the hitting time and place to a sphere or spherical shell for Brownian motion with a drift, Stat. Probab. Lett. 42 (1999) 367–373. Gerardo Hernandez-del-Valle: Department of Statistics, Columbia University, 1255 Amsterdam Ave. Room 1005, New York, N.Y 10027, U.S.A. E-mail address: [email protected]


Serials Publications www.serialspublications.com

INTRADAY EMPIRICAL ANALYSIS OF ELECTRICITY PRICE BEHAVIOUR ECKHARD PLATEN AND JASON WEST Abstract. This paper proposes an approach to the intraday analysis of the dynamics of electricity prices. The growth optimal portfolio (GOP) is used as a reference unit in a continuous financial electricity price model. A diversified global portfolio in the form a market capitalisation weighted index approximates the GOP. The GOP, measured in units of electricity, is normalised and then modelled as a time transformed square root process of dimension four. The dynamics of the resulting process is empirically verified. Intraday spot electricity prices from the US and Australian markets are used for this analysis. The empirical findings identify a simple but realistic model for examining the volatile behaviour of electricity prices. The proposed model reflects the historical price evolution reasonably well by using only a few robust and readily observable parameters. The evolution of the transformed time is modelled via a rapidly evolving market activity. A periodic, ergodic process with deterministic volatility is used to model market activity.

1. Introduction The dynamics of commodity prices plays a central role in valuing corresponding financial contingent claims. Their behaviour also plays a role in the evaluation of capital investments to produce a commodity. However, the main difficulty in implementing and testing commodity price models is that the typical state variables of these models do not seem to be directly observable. Employing a straightforward extension of the techniques developed under the continuous time Black and Scholes [2] and Merton [13] models and their extensions, has enabled researchers and practitioners to approximate the value of financial and real commodity contingent claims. Schwartz [16] employed three models that accounted for the mean reverting nature of commodity prices in differing ways. Interesting results in modelling commodity prices have also been obtained by Gibson and Schwartz [8], Cortazar and Schwartz [6] and Schwartz and Smith [17], using models similar to the Vasicek [18] interest rate framework. These models engage in examining the longer term dynamics of commodities however they do not provide a robust model for the short term dynamics of commodity prices, particularly the intraday dynamics. Received 2011-10-28; Communicated by the editors. 2000 Mathematics Subject Classification. Primary 91B26; Secondary 60H10. Key words and phrases. Intraday analysis, electricity price model, growth optimal portfolio, market activity. 721

722

ECKHARD PLATEN AND JASON WEST

The highly volatile nature of the behaviour of electricity prices suggests that if a market model can capture the dynamics of this erratic behaviour, then many other, more “well-behaved” commodities can also be modelled in a similar way using this approach. The application of existing commodity valuation models to electricity as a commodity has been shown to be unsatisfactory [5]. A number of alternative methods for modelling electricity have therefore been suggested. For instance, models that incorporate time varying volatility and price jumps have been employed in empirical settings [7]. More generally, modelling the most predictable components of electricity has been conducted [12]. However, each of these approaches ignores some important component in the modelling, such as jumps or stochastic volatility. Our methodology exploits the notion of the growth optimal portfolio (GOP), which maximises expected logarithmic utility. It was originally introduced by Kelly (1956) and later extended and applied in [11], [1], [14], and [4]. Under the standard risk neutral framework, the GOP coincides with the numeraire portfolio [11], which converts prices, when expressed in units of the GOP, into martingales under the real world probability measure. In a more general continuous time setting, Platen [14] demonstrates that when prices are denominated by the GOP, they become supermartingales. A diffusion model for the GOP is derived in [15] that is applied in the following analysis. Using a similar framework, [3] empirically constructed and modelled an intraday GOP denominated in U SD called the Market Capitalisation Weighted Index (MCI), with five minute observation intervals. This index is used as an approximation for the GOP in this paper. Under this framework we construct a market activity model for electricity prices using high-frequency trading data for the US and Australian electricity markets. Section 2 discusses the data set. Section 3 presents a framework for the GOP denominated in units of intraday electricity prices. Section 4 discusses the empirical analysis of intraday market activity and commodity price behaviour.

2. The Behavioural Characteristics of Electricity 2.1. Electricity Market Data. Electricity prices display a high degree of mean reversion and are subject to significant intraday, day of the week and seasonal patterns. Some are censored from above, but not from below. In the short term, these prices seem to be determined by the level of demand while in the long term, they appear to fluctuate around the cost of production, which emphasises the presence of a mean reversion property. The effect of seasonal variations in electricity prices has been well documented, see [5], [10] and [12]. Local air temperatures heavily influence the demand for electricity which, if excessive, places significant burdens on the electricity generators for the immediate production of electricity. The electricity market is subject to distribution and transmission constraints, such that once fully constrained, the marginal cost of transmission can become practically infinite. In some markets, such as the US and Australia, price caps have been instituted to combat this possible situation. There is also the capacity for slightly negative prices, which occur as a consequence of an inability to freely dispose excess electricity.

INTRADAY ELECTRICITY PRICE BEHAVIOUR

723

350

300

Price ($USD/MWh)

250

200

150

100

50

0 2000

2000.5

2001

2001.5

Year

Figure 1. NEPOOL spot electricity prices EPU SD (t) Jan 2000 - Jul 2001. The market for electricity is an auction where generators and distributors submit bids encompassing volume and price information. Extremes in temperatures coupled with outages in generation or transmission induce price spikes to occur at random points in time. These price spikes are characteristic for electricity as a traded commodity, which needs to be produced on demand due to limited storage capability. Finally, it appears that the volatility of observed electricity prices tends to rise more with positive shocks than with negative shocks, a phenomenon referred to as the inverse leverage effect [10]. This property will be naturally incorporated in our modelling. 2.2. Data Analysis. The worldwide trend towards deregulating electricity markets is an attempt to promote competition at each stage of the electricity production and supply chain. In the US and Australian wholesale electricity markets, the electricity output from generators is pooled and then scheduled to meet electricity demand. Trading in electricity is conducted in a spot market, which allows instantaneous matching of supply against demand. Generators offer to supply the market with different amounts of energy at particular prices. Each market boasts its own set of regulations and trading conventions. For instance, in the US, the New England Power Pool (NEPOOL), administered by the independent service operator (ISO) New England, dispatches electricity at an average generation price aggregated across all power providers every hour. In Australia until 2009, the National Electricity Market Management Company (NEMMCO) was the independent regulator responsible for the dispatch of electricity across four Australian states. The NEMMCO is now known as the Australian Energy Market Operator (AEMO). Dispatch instructions are sent to each generator at five minute intervals to schedule the amount of power to be produced and

724


350

300

Price ($AUD/MWh)

250

200

150

100

50

0 2000

2000.5

2001

2001.5

Year

Figure 2. NSW spot electricity prices EPAU D (t) Jan 2000 - Jul 2001.

the six dispatch prices recorded during each half hour period are averaged to determine the spot price. Prices are calculated for dispatch intervals in each region in Australia, of which New South Wales (NSW) is the largest. In this analysis, we will refer to US prices as the NEPOOL electricity price exclusively expressed in US dollars (U SD), and to Australian prices as the NSW electricity price exclusively expressed in Australian dollars (AU D). For this study, hourly spot prices have been obtained for the NEPOOL and half-hourly spot electricity prices for NSW were obtained. Although electricity prices have been deregulated in both countries since 1998, there appears to be some structural inhibitors in the prices prior to 2000, resulting in a clearly defined and seemingly artificial level to which prices revert. This hints at the existence of some residual regulatory effects within each market, and therefore we ignore spot electricity price data prior to January 2000. We denote the spot electricity price at time t by EPd (t), where we set d = U SD when we consider the NEPOOL spot electricity price, which is denominated in U SD. Similarly, we denote by EPAU D the NSW spot electricity price at time t, which is denominated in AU D. We conduct our modelling of electricity prices based on spot price data for the NEPOOL and NSW electricity markets from January 2000 to July 2001, which amounts to 13128 spot electricity prices for the NEPOOL and 26256 spot electricity prices for NSW. We shall ignore slightly negative prices as they are of insignificant magnitude and occur at infrequent and irregular times. Figures 1 and 2 show spot electricity prices EPUSD (t) and EPAU D (t) for the eighteen month period for each market. The prices are clearly fluctuating, featuring numerous spikes based upon an underlying time dependent reference level. Some prices during this period reached levels in excess of $5000/MWh. Higher prices appear to be aligned


725

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1 2000

2000.5

2001

2001.5

Year

Figure 3. Market Capitalisation Weighted Index (MCI) in U SD, Jan 2000 - Jul 2001. 3.1

3

2.9

2.8

2.7

2.6

2.5

2.4

2.3

2.2 2000

2000.5

2001

2001.5

Year

Figure 4. Market Capitalisation Weighted Index (MCI) in AU D, Jan 2000 - Jul 2001. with the extremes in temperatures, primarily during extreme winter and summer conditions. Following the approach by Platen [14] we use the market capitalisation-weighted world stock index (MCI) as a diversified portfolio of stocks and as a proxy for the growth optimal portfolio (GOP). In Figures 3 and 4 we display the intraday MCI (MCI) denominated in U SD and AU D, which is denoted at time t by Sd (t) with

726


d ∈ {U SD, AU D}, as constructed in [3]. The stock markets aggregated in this MCI account for more than 95 percent of the total world market capitalisation. The intraday MCI is formed as a self financing portfolio using 34 local stock market indices taken from almost all financial markets throughout the world, weighted by market capitalisation and rebalanced according to a strict set of rules. This index generally mimics the Morgan Stanley Capital Growth World Index (MSCI) over the period, but it provides observed values at a five minute frequency. In fact, the MCI outperforms the MSCI over the time period used in this analysis. To be consistent with the observation frequency, for NEPOOL data we arithmetically average twelve consecutive five minute intervals to compute an hourly average for the MCI in U SD with values shown in Figure 3. For NSW data, we arithmetically average six consecutive five minute intervals to compute the half hourly average for the MCI in AU D shown in Figure 4. This method of averaging the index data over time is employed since the hourly NEPOOL and half hourly NSW spot prices for electricity are similarly obtained by averaging the dispatch prices over consecutive five minute intervals. These are the highest frequencies at which electricity price data were available for the two markets. The time steps for both the MCI values and the spot electricity prices are synchronised by this method. We now denominate the MCI in units of the ith spot electricity price, i ∈ EP = {EPUSD , EPAUD }, at time t ∈ [0, T ] by setting (MCI)

SEPd

(MCI)

(t) =

Sd (t) , EPd (t)

(2.1)

for d ∈ {U SD, AU D}. Since the observation intervals for the processes are synchronised, the values for (MCI) Si (t), i ∈ EP , at the observation times t ∈ [0, T ] are easily obtained. It is (MCI) appropriate to normalise the MCI Si (t) at time t when denominated in units of the ith electricity price. For this, we introduce the ith normalisation function α¯i (t) = ξi eηi t ,

(2.2)

for i ∈ EP and t ∈ [0, T ], where ξi > 0 is a normalisation factor and ηi > 0 is the growth rate of the ith normalisation function. Thus we introduce the ith normalised MCI Yi (t) at time t for the corresponding electricity price in the form (MCI)

Yi (t) =

Si

(t) , α¯i (t)

(2.3)

for i ∈ EP and t ∈ [0, T ]. This representation is a normalised MCI time series which will assist with the formulations through the remainder of the paper. We choose ξEPU SD = 28 and ξEPAU D = 63 as the appropriate normalisation factors. The values that we use for the growth rates are ηEPU SD = 0.0127 and ηEPAU D = 0.0257, which follows from a detailed analysis of the behaviour of electricity prices over the long term and will be given in Section 3. However, it transpires that for the relatively short time period of 1.5 years that we consider, the particular value of the growth rate is not really important. The inclusion of the above normalisation function accommodates both the long term growth rate


727

0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0 2000

2000.5

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2001.5

Year

Figure 5. Normalised MCI in denomination of NEPOOL electricity prices YEPU SD (t). 0.018

0.016

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Figure 6. Normalised MCI in denomination of NSW electricity prices YEPAU D (t).

of the MCI in U SD or AU D, and the growth implicit in electricity prices related to inflation. The evolution of the normalised MCI Yi (t), i ∈ EP , at time t ∈ [0, T ] is shown in Figure 5 for the denomination in NEPOOL electricity prices, and in Figure 6 for the denomination in NSW electricity prices. The magnitude of the fluctuations depends on the average level of the normalised MCI Yi (t), i ∈ EP . Periods of high values correspond to periods of large

728


fluctuations and periods of low values correspond to periods of small fluctuations. Taking the square root increases fluctuations when the level of the normalised index is low and decreases fluctuations when the normalised index is high. Therefore, to alter p the fluctuations to obtain prices of a similar size, we consider the square root Yi (t) of the normalised MCI for i ∈ EP . This allows the fluctuations to be reasonably similar for both high and low levels. The quadratic variation is the sum of the squares of the increments of the process in question when the time discretisation step size vanishes. Its slope provides information about its fluctua√ tions. Figures 7 and 8 illustrate the quadratic variation h Yi it of the square root of the normalised MCI for units of the electricity price for NEPOOL and NSW spot prices, respectively. The graphs in Figures 7 and 8 appear to be relatively smooth, apart from some small periods of major changes in the slope, particularly for the NEPOOL data. They are roughly proportional to a quarter of the time that has elapsed. However, we note that the slope changes in a possibly annually oscillating fashion. The quadratic variation related to NEPOOL prices illustrated in Figure 7, demonstrates the feature of seasonality in a more obvious way than the seasonality evident from the quadratic variation of NSW prices. The New England region in the US experiences far greater extremes in air temperatures than NSW in Australia, which explains the greater degree of seasonality in NEPOOL electricity price fluctuations relative to those of NSW electricity prices. The slopes of the curves in Figures 7 and 8 are greater during periods of higher average fluctuations, than the slope is during periods with significantly more stable behaviour, and this property exists throughout the data p set. Linear regression shows an R2 value of 0.9584 for the quadraticpvariation h YEPU SD (t)it , and an R2 value of 0.9654 for the quadratic variation h YEPAU D (t)it , confirming that it is reasonable to assume linearity, as first approximation. One could make the growth rate ηi time dependent and seasonal, which would allow the curves in Figures 7 and 8 to become almost perfectly straight lines. However, this would need a much longer data set or more detailed modelling to be reasonably founded. 3. A Numeraire Approach According to [14] the growth optimal portfolio (GOP) can be interpreted as a diversified accumulation index, representative of the best performing long term investment policy. It maximises the expected logarithm of terminal wealth and thus the expected portfolio growth rate. In the very long term, the growth optimal policy almost surely provides higher wealth than alternative strategies [11]. Consider a sequence of market models indexed by a number d ∈ {1, 2, ...}. For a given value of d the corresponding financial market model comprises d + 1 primary securities. For a given strategy δ, it is convenient to introduce the jth (j) (j) proportion πδ of the value Sδ of the corresponding strictly positive portfolio that is invested at time t ∈ [0, T ] in the jth primary security account, j ∈ {0, 1, ..., d} and d ∈ {1, 2, ...}. This proportion or fraction is given by the expression (j)

πδ (t) = δ (δ) (t)

(j)

Sd (t) (δ)

Sd (t)

,

(3.1)


729

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 2000

2000.5

2001

2001.5

Year

p Figure 7. Quadratic variation h YEPU SD it of the square root of the normalised MCI for NEPOOL electricity prices. 0.4

0.35

Quadratic Variation

0.3

0.25

0.2

0.15

0.1

0.05

0 2000

2000.5

2001

2001.5

Year

p Figure 8. Quadratic variation h YEPAU D it of the square root of the normalised MCI for NSW electricity prices. for t ∈ [0, T ] and j ∈ {0, 1, ..., d}. Note that the proportions always sum to one, that is d X (j) πδ (t) = 1, (3.2) j=0

for t ∈ [0, T ].

730


In order to approximate the GOP using a diversified portfolio we formulate the following definitions. Definition 3.1. A strictly positive portfolio process S (δ) is called a diversified portfolio (DP) if finite constants K1 > 0, K2 > 0 and K3 ∈ {1, 2, ...} exist, independent of d, such that K1 (j) |πδ | ≤ 1 +K , (3.3) d2 2 almost surely for all j ∈ {0, 1, ..., d}, d ∈ {K3 , K3 + 1, ...} and t ∈ [0, T ]. (j)

This means that the proportion πδ of the value of a DP, which is invested 1 in the jth primary security account, need to decrease slightly faster than d− 2 as d → ∞. This is, for instance, the case if equal proportions are used. The key pricing features of the financial market have been characterised via the GOP where a model for the dynamics of the GOP has been suggested. In this paper we apply these results to the case of electricity which is a commodity. Instead of referring to the GOP in units of dollars, we express it in units of electricity and analyse its dynamics. More specifically, we interpret the MCI when expressed at (MCI) time t in units of electricity Si (t), i ∈ EP , as being the GOP that is measured in units of electricity. This has consequences for the structure of the dynamics of (MCI) the value Si (t). When we express a given portfolio S (δ) (t) in units of the GOP, then we call the ratio S (δ) (t) Sˆ(δ) (t) = (δ ) , (3.4) S ∗ (t) the corresponding benchmarked portfolio. By application of the Itô formula and using 3.3 it follows that the benchmarked portfolio Sˆ(δ) (t) satisfies the SDE dSˆ(δ) (t) = −Sˆ(δ) with jth specific volatility

d X d X

(j)

πδ (t)σ j,k (t)dW k (t)

(3.5)

k=1 j=0

σ j,k (t) = bj,k (t) − θk (t)

(3.6)

for t ∈ [0, T ], j ∈ {0, 1, ..., d} and k ∈ {1, 2, ..., d}. This formulation permits specific volatility to be negative under some conditions. This allows us to introduce the kth total specific volatility σ ˆ k (t) =

d X j=0

|σ j,k (t)|

(3.7)

for t ∈ [0, T ] and k ∈ {1, 2, ..., d}. (MCI) From 3.5 the GOP value Si (t) defined in 2.1 satisfies the stochastic differential equation (SDE) (MCI)

dSi

(MCI)

(t) = Si

(t) [ri (t)dt + θi (t)(θi (t)dt + dWi (t))] ,

(3.8)

for i ∈ EP and t ∈ [0, T ]. Here Wi = {Wi (t), t ∈ [0, T ]} is a standard Wiener process on a given probability space. This representation is standard for contingent


731

claim modelling in financial markets. The volatility θi (t) is the market price for risk with respect to Wi . If the electricity price is interpreted as a currency, then ri (t) refers to its interest rate. In the case of the given commodity, we call this the electricity short rate, which is technically analogous as the short term interest rate is for currencies. In the SDE (3.8) the risk premium is the square of the volatility. Definition 3.2. A benchmark model is called regular if there exist finite constants K3 and K4 , independent of d, such that E((ˆ σ k (t))2 ) < K4

(3.9)

for all t ∈ [0, T ], k ∈ {1, 2, ..., d} and d ∈ {K3 , K3 + 1, ...}. This is a property that can be assumed to be represented by the world stock market consisting of all stocks traded on existing exchanges. The difference between the logarithms of the GOP S (δ∗ ) (t) and a given strictly positive portfolio S (δ) (t) satisfies the SDE d

k=1

with tracking rate Rδd (t) =

d

X X (j) 1 d Rδ (t)dt − πδ (t)σ j,k (t)dW k (t) (3.10) 2 j=0

d(log(S (δ∗ ) (t)) − log(S (δ) (t))) =

d X d X (j) ( πδ (t)σ j,k (t))2

(3.11)

k=1 j=0

for all t ∈ [0, T ]. The tracking rate equals the squared diffusion coefficient of the SDE 3.10. It can be interpreted as a measure of the distance between a given portfolio S (δ) (t) and the GOP S (δ∗ ) (t) at time t ∈ [0, T ]. Definition 3.3. For an increasing number d of risky primary security accounts we call a strictly positive portfolio S (δ) an approximate GOP if the corresponding sequence of tracking rates (Rδd (t))d∈{1,2,...} vanishes in probability, that is for each ǫ > 0 we have lim P (Rδd (t)) > ǫ) = 0 (3.12) d→∞

for all t ∈ [0, T ]. Under the above assumptions we propose the following limit theorem. Theorem 3.4. For a regular benchmark model a diversified portfolio is an approximate GOP. Proof. We estimate by using 3.11, 3.3 and 3.9 for a DP S (δ) in a regular benchmark model its expected tracking rate. That is, edδ (t) = E(Rδd (t)) ≤ ≤

d X

k=1

d X (j)(t) E(( |πδ ||σ j,k (t)|)2 )

(3.13)

j=0

d X (K1 )2 ( 1+2K2 K4 ) d

(3.14)

k=1

≤ (K1 )2 K4 d−2K2

(3.15)

732


for t ∈ [0, T ] where d ∈ {K3 , K3 + 1, ...}. Consequently, since K2 > 0 it follows by the Markov inequality for any given ǫ > 0 that 1 lim P (Rδd (t)) > ǫ) ≤ lim edδ (t) = 0 (3.16) d→∞ d→∞ ǫ for all t ∈ [0, T ]. This proves by Definition 3.3 the Proposition 3.4. This result allows us to conclude that a world stock portfolio that is a DP approximates the GOP and justifies the use of the diversified portfolio to benchmark electricity prices in this analysis as discussed in Section 2.2. (MCI)

3.1. Discounted GOP. Let us discount the GOP value Si (t) in ith electricity units at time t, see (3.8), by the electricity price savings account value Z t (0) Si (t) = exp ri (s)ds , (3.17) 0

at time t ∈ [0, T ] for i ∈ EP , where electricity is theoretically accrued at the (MCI) electricity short rate ri (t). Let us discount the GOP value Si (t), see (3.8), at (0) time t by the electricity savings account Si (t). Then the discounted GOP (MCI)

S (t) (MCI) S¯i (t) = i (0) , Si (t)

(3.18)

satisfies by application of the Itô formula, (3.8) and (3.17) the SDE (MCI)

dS¯i

(MCI) (t) = S¯i (t)θi (t)(θi (t)dt + dWi (t)),

(3.19)

for t ∈ [0, T ] and i ∈ EP . By (3.19), using the discounted GOP drift (MCI)

αi (t) = S¯i

(t)(θi (t))2 ,

as a parameter process leads to a GOP volatility at time t of the form s αi (t) θi (t) = . (MCI) ¯ S (t)

(3.20)

(3.21)

i

We consider a rather short time period. Therefore we assume, for simplicity, ri (t) = 0 for t ∈ [0, T ] and i ∈ EP . Similar to the normalisation function α¯i (t) in (2.2) we model the discounted GOP drift in the form αi (t) = α ¯ i (t)mi (t),

(3.22)

for i ∈ EP and t ∈ [0, T ]. Here mi = {mi (t), t ∈ [0, T ]} denotes the nonnegative ith market activity process that has an average value of one for i ∈ EP that will be further specified in Section 3.2. The numeraire-denominated price processes are therefore normalised so that the GOP is aligned with electricity price processes. To derive a value for the ith net growth rate ηi , several concerns must be addressed. Electricity, as a commodity, is a security that theoretically has a time value. An income can be obtained from lending this commodity since there is value in being able to make the commodity available over certain periods. We avoid particular detailed assumptions regarding the nature of the growth rate or


733

the electricity short rate. These appear as natural quantities in our setup. The notion of a net convenience yield, separated into gross convenience and cost of carry in our setting appears to be unnecessary. 3.2. Market Activity. Based on the market activity mi (t), the ith market activity time ψi = {ψi (t), t ∈ [0, T ]} can be defined as Z t ψi (t) = mi (s)ds, (3.23) 0

for t ∈ [0, T ] and i ∈ EP . In our case t = 0 years corresponds to the starting date of our sample 01/01/2000 00:00:00 for Greenwich Mean Time (GMT). Furthermore, we ensure that on average the market activity time scale elapses approximately as fast as physical time. Approximately ψi (T ) ≈ 1.5 will turn out to be equivalent to 30/06/2001 00:00:00 GMT, the terminal date of both sets of data available. To be precise, we assume that asymptotically we have in our model the property that 1 E (ψi (T )) = 1. T →∞ T lim

(3.24)

The normalised MCI Yi = {Yi,ψ , ψ ∈ [t, ψi (T )]} can be conveniently expressed in market activity time and is then obtained as Yi,ψi (t) = Yi (t) =

(MCI) (MCI) S¯i (t)mi (t) S¯ (t) = i , αi (t) α(t) ¯

(3.25)

for i ∈ EP . Let us introduce the ith market activity in ith activity time, that is mi,ψi (t) = mi (t),

(3.26)

for t ∈ [0, ψi (T )] and i ∈ EP . It is has been shown via the Itô formula, and using (3.19) and (3.22) in [15], that (3.25) in ith market activity time, satisfies the SDE p 1 Yi,ψ dYi,ψ = ηi ( − )dψ + Yi,ψ dWi,ψ , (3.27) ηi mi,ψ for i ∈ EP and ψ ∈ [0, ψi (T )], where

dWi,ψi (t) =

p mi (t)dWi (t),

(3.28)

for t ∈ [0, T ]. In the case of mi,ψ = 1, the normalised MCI in market activity time, given in (3.25), is a square root process of dimension four. In this case, the solution of (3.27) has a long term mean of η1i and a speed of adjustment parameter ηi . The only parameter that is then relevant in (3.27) is the growth rate ηi , which turns out to be the key growth parameter for the price of electricity. If we consider the square root of the normalised MCI, then by (3.27) and application of the Itˆ o formula, it evolves according to the SDE ! p p 3 ηi Yi,ψi 1 p d( Yi,ψ ) = − dψ + dWi,ψ , (3.29) 2 mi,ψ 2 8 Yi,ψ

for i ∈ EP and ψ ∈ [0, ψi (T )]. It is crucial to note that the diffusion coefficient in (3.29) is constant. Therefore, we obtain in market activity time the quadratic

734

variation of


√ Yi in the form

p ψ h Yi iψ = , (3.30) 4 for i ∈ EP and ψ ∈ [0, ψi (T )]. Relation (3.30) holds under general circumstances (MCI) since no major restrictive assumptions on the actual dynamics of S¯i have been made. √ From the market activity time ψi given in (3.23) and the quadratic variation of Yi in (3.30), we formulate the following result. Corollary 3.5. The ith market activity can be calculated as the time derivative √ dψi (t) dh Yi it mi (t) = =4 , (3.31) dt dt for i ∈ EP and t ∈ [0, T ]. This shows that the discounted GOP drift is equal to four times the slope of the quadratic variation of the square root of the discounted GOP. This implies that market activity is directly observable as a time derivative. One needs only to measure the slope of the quadratic variation of the square root of the normalised MCI. (MCI) For the intraday observed normalised MCI Yi (t), in units of NEPOOL and NSW electricity, the quadratic variation of its square root has been shown in Figures 7 and 8, respectively. The slope of these graphs at a given time provides by (3.31), a quarter of the corresponding market activity. This allows us to calculate the market activity directly. We simply calculate the numerical derivative corresponding to (3.31) using hourly time steps for the NEPOOL electricity prices and half hourly time steps for NSW electricity prices, and multiply this by four. In Figure 9 we plot the resulting market activity for the NEPOOL series. The market activity fluctuates over a wide range. Therefore, we show in Figure 10 the logarithm log(mi (t)) of this derivative over a few weeks in April/May 2000 for the NEPOOL series. The NSW electricity series results are similar. It appears that the observed market activity processes show some seasonal patterns and display the characteristic of reverting quickly back to a reference level. It also distinctly shows periods of market inactivity, which typically occur at night when minimal variation in electricity usage is usually experienced. The quadratic variation of the logarithm of market activity hlog(mi )it , calculated as a rolling time average over the scenarios, is shown in Figures 11 and 12, which demonstrates that there is minimal seasonal pattern emerging from market activity over the long term. Note that the average slope of the quadratic variation in both Figure 11 and Figure 12 is close to one. We show in Figures 13 and 14 the quadratic variation of the logarithm of market activity examined at a higher resolution for both the NEPOOL and NSW electricity markets. There is, of course, a distinct intraday seasonal pattern which will be discussed in Section 4 below. Figures 13 and 14 illustrate that nights are characterised by a plateau, which means low activity volatility. Notably, despite the seasonal pattern, the graphs in Figures 13 and 14 appear to be roughly linear for the periods when the market is actively trading. The deseasonalisation of the


735

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30

25

20

15

10

5

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Figure 9. Market activity mEPU SD (t) for NEPOOL electricity prices. 4

2

0

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-4

-6

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-12

-14 04/15/00

04/22/00

04/29/00

05/06/00

Date

Figure 10. Logarithm log(mEPU SD (t)) of NEPOOL market activity for April - May 2000. market activity mi (t) requires a two stage procedure. The first step deseasonalises the average of the market activity and the second its volatility. This will be outlined below. 4. Model for Market Activity 4.1. Activity Volatility. Figures 11 and 12 suggest that due to the approximate linearity of the quadratic variation hlog(mi )it , market activity is likely to have

736


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1.4

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Figure 11. Quadratic variation of the logarithm of market activity for NEPOOL hlog(mEPU SD )it . 1.6

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1.2

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Figure 12. Quadratic variation of the logarithm of market activity for NSW hlog(mEPAU D )it . multiplicative noise. In this paper, we propose a linear mean reverting model for the market activity mi (t) of electricity that accounts for this feature. It is given by the SDE ¯ i (t), dmi (t) = κi βi2 (t)(m ¯ i (t) − mi (t))dt + βi (t)mi (t)dW (4.1) for i ∈ EP and t ∈ [0, T ], with deterministic speed of adjustment parameter κi > 0, reference market activity m ¯ i (t) ≥ 0 and activity volatility βi (t) > 0. Here,


737

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0.06

0.05

0.04

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0.01

0 04/15/00

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Figure 13. Quadratic variation of the logarithm of market activity for NEPOOL hlog(mEPU SD )it , April - May 2000. 0.09

0.08

0.07

Quadratic Variation

0.06

0.05

0.04

0.03

0.02

0.01

0 04/15/00

04/22/00 Date

04/29/00

05/06/00

Figure 14. Quadratic variation of the logarithm of market activity for NSW hlog(mEPAU D )it , April - May 2000. ¯ = {W ¯ i (t), t ∈ [0, T ]} for i ∈ EP is an independent standard Wiener process. W The reference market activity m ¯ i (t) and the activity volatility βi (t) are assumed to exhibit some deterministic seasonal pattern. Let us also introduce the expected market activity m ˆ i (t) at time t ∈ [0, T ] as the expectation m ˆ i (t) = E(mi (t)) (4.2)

738


6

5

4

3

2

1

0 Sat

Sun

Mon

Tue

Wed

Thu

Fri

Day

Figure 15. Weekly seasonal pattern of expected market activity m ˆ EPU SD (t) for NEPOOL. for t ∈ [0, T ] and i ∈ EP . To deseasonalise mi (t), we estimate the expected market activity m ˆ i (t) by simply using the Law of Large Numbers for each observation time of the week during the full observation period for NEPOOL and NSW data, respectively. Figures 15 and 16 illustrate the observed weekly pattern in the average market activity m ˆ i (t) for i ∈ EP . It indicates that, on average, market activity reaches two distinct peaks during weekdays and one peak on weekends for both NEPOOL and NSW electricity markets. This pattern is very similar over all seasons of the year. We also see that nights display almost no market activity, as would be expected. 4.2. Activity Volatility Time. The deterministic seasonal ith activity volatility βi (t) allows us to introduce the ith activity volatility time τi = {τi (t), t ∈ [0, T ]} as Z t

τi (t) = hlog(mi )it =

(βi (u))2 du

(4.3)

0

for t ∈ [0, T ] and i ∈ EP and. Again, as with market activity time, the activity volatility time (4.3) requires initialisation, which we have simply set τi (0) = 0. Figures 17 and 18 show the average weekly activity volatility series. Public holidays were counted as Sundays since the behaviour of electricity prices on public holidays closely resembles, on average, the behaviour of electricity prices on Sundays. This shows that the average activity volatility for weekdays is different from the average activity volatility on weekends. It seems that electricity demand and prices appear, in general, to be more predictable on weekdays than on weekends, as reported for the case of Australia in [5]. This feature results in a more stable activity volatility pattern emerging during the week, which is illustrated in Figures 17 and 18.


739

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3

2.5

2

1.5

1

0.5

0 Sat

Sun

Mon

Tue

Wed

Thu

Fri

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Figure 16. Weekly seasonal pattern of expected market activity m ˆ EPAU D (t) for NSW. 4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 Sat

Sun

Mon

Tue

Wed

Thu

Fri

Day

Figure 17. Average weekly pattern of activity volatility βU SD (t) for NEPOOL. To extract the reference market activity m ¯ i (t) it is useful to consider the market activity in activity volatility time τi (t). When we denote the market activity, reference market activity and average market activity in activity volatility time, by mi,τi (t) = mi (t), m ¯ i,τi (t) = m ¯ i (t) and m ˆ i,τi (t) = m ˆ i (t) respectively, we obtain by (4.1) and (4.3) the SDE ¯ i,τ , dmi,τ = κi (m ¯ i,τ − mi,τ )dτ + mi,τ dW

(4.4)

740


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2.5

2

1.5

1

0.5

0 Sat

Sun

Mon

Tue

Wed

Thu

Fri

Day

Figure 18. Average weekly pattern of activity volatility βAU D (t) for NSW. which progresses in units of τ ∈ [0, τi (T )], where

¯ i,τ (t) = βi (t)dW ¯ i (t), dW i

(4.5)

for t ∈ [0, T ] and i ∈ EP . By taking expectations on both sides of the SDE (4.4) we obtain by (4.2) the ordinary differential equation dm ˆ i,τ = κi (m ¯ i,τ − m ˆ i,τ )dτ,

(4.6)

for t ∈ [0, τi (T )] and thus the reference level for the market activity in the form m ¯ i,τ =

ˆ i,τ 1 dm +m ˆ i,τ , κi dτ

(4.7)

for τ ∈ [0, τi (T )] and i ∈ EP . Below we will estimate the speed of adjustment parameter κi for i ∈ EP . By (4.7) this gives, together with the average market activity estimated in Figures 15 and 16, an estimate for the reference level m ¯ i (t) = m ¯ i,τi (t) . It is clear from (4.7) that we require the derivative of the expected market activity with respect to activity volatility time. Provided that expected market activity is relatively smooth, the reference market activity can be calculated. From (4.4) and (4.6) and by application of the Itô formula, the logarithm of ith market activity log(mi,τ ) in ith activity volatility time satisfies the SDE 1 dm ˆ i,τ κi m ˆ i,τ 1 ¯ i,τ , d log(mi,τ ) = + − κi − dτ + dW (4.8) mi,τ dτ mi,τ 2 for τ ∈ [0, τi (T )] and i ∈ EP . We therefore have a nonlinear drift term and a constant diffusion term for the SDE of the logarithm of market activity log(mi,τ ) in activity volatility time.


741

0.025

Quadratic Variation

0.02

0.015

0.01

0.005

0 04/15/00

04/22/00 Date

04/29/00

Figure 19. Quadratic variation of the logarithm of market activity hlog(mEPU SD )iτ for NEPOOL in activity volatility time.

0.02

Quadratic Variation

0.015

0.01

0.005

0 04/15/00

04/22/00 Date

04/29/00

05/06/00

Figure 20. Quadratic variation of the logarithm of market activity hlog(mEPAU D )iτ for NSW in activity volatility time.

The quadratic variation of the logarithm of market activity for the NEPOOL and NSW electricity markets is shown in Figures 19 and 20 in the respective activity volatility time. The higher frequency of NSW data compared with the NEPOOL data translates into a smoother quadratic variation curve as is evident in these figures. The relative linearity of these curves supports the presence of

742


multiplicative noise, which is assumed in the SDE (4.1) for market activity. Theoretically, this leads by (4.8) to hlog(mi )iτ = τ . In this paper we interpret the average weekly activity volatility, shown in Figures 17 and 18, as activity volatility βi (t) for each week of the period considered. 4.3. Estimation of Speed of Adjustment. The only parameter in (4.1) remaining to be estimated is the speed of adjustment parameter κi , i ∈ EP , that controls the average strength of mean reversion. Figure 21 shows the histogram of the logarithm li,τ = log(mi,τ ) of the market activity for the NEPOOL electricity market, i ∈ EP . As shown, a concentration of negative spikes at log(mEPU SD ,τ ) ≈ −14 exists in Figure 21 respectively, due to the opening and closing effects that occur around the relatively benign activity level typical for each night. As in [3], we shall exclude this distortion from our analysis by forming a restricted log-likelihood function, where only the values for log(mEPU SD ,τ ) > −10 and log(mEPAU D ,τ ) > −15 for each histogram, respectively, are considered. We require a robust estimation technique to estimate κi . Under the simplifying assumption that m ¯ i,τ = 1, i ∈ EP , the market activity process can be shown to have as a stationary density, an inverse gamma density with a scale parameter of 1. The stationary transition density p¯(l; κi ) of the logarithm of market activity in activity volatility time can therefore be written as (κi )κi p¯(l; κi ) = exp −κi e−l e−l(κi −1) , (4.9) Γ(κi )

for i ∈ EP . A maximum likelihood technique with the above restriction was applied. A plot of the estimated probability density function of log(mi,τ ), based on the resulting maximum likelihood estimate of κi , is shown for the US data in Figures 21. The Australian results are largely similar. We estimated κEPU SD to be about κ ˆ EPU SD = 11.4 with a 99% confidence interval of (10.8, 14.5), and κEPAU D to be about κ ˆ EPAU D = 25.2 with a 99% confidence interval of (23.8, 28.5). This translates into an expected reversion to the mean with a half life time following a shock of about 22 days for NEPOOL electricity prices and about 10 days for NSW electricity prices. Maximum likelihood estimates for a two parameter G(l; κi , m ¯ i ) inverse gamma density function were also obtained. For both the NEPOOL and NSW data, the reference level parameter m ¯ i yielded a value of approximately 1, with estimates for the shape parameter κi converging close to the above values obtained using a single parameter estimate for the likelihood function of the inverse gamma density, i ∈ EP . This further strengthens the argument for the applicability of the inverse gamma density function as an appropriate fit for the stationary density of the logarithm of market activity, from which a reliable estimate of the speed of adjustment parameter can be obtained. In Section 3 it was shown that the normalised MCI, when observed in market activity time, resembles a square root process of dimension four. By using market activity time we know that the quadratic variation of its square root should be linear with a slope close to 0.25, see (3.30). This relationship is confirmed by performing a simple linear regression of the quadratic variation of the square root


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0.12

0.1

0.08

0.06

0.04

0.02

0 -15

-10

-5

0

5

10

Ln(m)

Figure 21. Histogram and estimated probability density function of log(mEPU SD (t)).

of the normalised MCI against market activity time. The slope coefficient for the NEPOOL electricity market is 0.2531 with corresponding R2 = 0.9584, and for the NSW electricity market, the slope coefficient is 0.2517 with corresponding R2 = 0.9985. This makes the above derived and calibrated model a largely accurate intraday description of electricity prices. Additionally, the corresponding market activity process in activity volatility time is largely shown to have the hypothesised dynamics, when the distorting effects that occur at the close and opening of trading are omitted. On the basis of the above detailed modelling and calibration, it is possible to price electricity derivatives which, however, is beyond the scope of this paper. 5. Conclusion We have examined the behaviour of an intraday world stock index when expressed in units of electricity prices for two distinct markets. We were able to construct a model for electricity prices that allows for both seasonalities and long term growth. The behaviour of the market activity of electricity prices was inferred through this analysis. A simple way of calculating market activity was subsequently demonstrated. The model for market activity yielded robust results for high frequency electricity price data. This suggests that similar models can possibly be employed for other types of commodities. Market activity is shown to contain seasonal patterns in both the drift and the diffusion term within a coherent format. We showed that the market activity can be modelled as a strongly mean reverting process. Furthermore, we confirmed that the normalised index closely follows a square root process of dimension four in market activity time. Using

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the techniques outlined in this paper, a method of obtaining derivative prices for electricity will be described in forthcoming work. Acknowledgment. The authors would like to thank Olsen & Associates, Professor Wolgang Breymann and Datastream as well as the New England Power Pool and the Australian Energy Market Operator for providing data that is relevant for this study. The authors also wish to thank an anonomous referee for insightful comments. References 1. Bajeux-Besnainou, I. and Portait, R.: The numeraire portfolio: A new perspective on financial theory, European Journal of Finance 3 (1997) 291–309. 2. Black, F. and Scholes, M.: The valuation of option contracts and corporate liabilities, Journal of Political Economy 81 (1973) 637–659. 3. Breymann, W., Kelly, L., and Platen, E.: Intraday empirical analysis and modelling of diversified world stock indices, QFRC Research Paper 125 University of Technology, Sydney (2004). 4. B¨ uhlmann, H. and Platen, E.: A discrete time benchmark approach for insurance and finance, ASTIN Bulletin 33(2) (2003). 5. Clewlow, L. and Strickland, C.: Energy Derivatives: Pricing and Risk Management, Lacima Publications, London, 2000. 6. Cortazar, G. and Schwartz, E.: The valuation of commodity contingent claims, Journal of Derivatives 1 (1994) 27–39. 7. Eydeland, A. and Geman, H.: Some fundamentals of electricity derivatives, Working Paper, Southern Company Energy Marketing and University Paris IX Dauphine and ESSEC (1998). 8. Gibson, R. and Schwartz, E.: Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance 45 (1990) 959–976. 9. Kelly, J. R.: A new interpretation of information rate, Bell Systems Technology Journal 35(1) (1956) 917–926. 10. Knittel, C. R. and Roberts, M. R.: An empirical examination of deregulated electricity prices, Power Working Paper, University of California Energy Institute (2001). 11. Long, J. B.: The numeraire portfolio, Journal of Financial Economics 26 (1990) 29–69. 12. Lucia, J. and Schwartz, E.: Electricity prices and power derivatives: Evidence from the Nordic Power Exchange, Review of Derivatives Research 5 (2002) 5–50. 13. Merton, R. C.: Theory of rational option pricing, Bell Journal of Economics and Management Science 4 (1973) 141–183. 14. Platen, E.: Arbitrage in continuous complete markets, Advances in Applied Probability 34 (2002) 540–558. 15. Platen, E.: Diversified portfolios in a benchmark framework, QFRC Research Paper 87 University of Technology, Sydney (2003). 16. Schwartz, E.: The stochastic behaviour of commodity prices: Implications for valuation and hedging, Journal of Finance 52(3) (1997) 923–973. 17. Schwartz, E. and Smith, J.: Short-term variations and long-term dynamics in commodity prices, Management Science 46(7) (2000) 893–911. 18. Vasicek, O.: An equilibrium characterisation of the term structure, Journal of Financial Economics 5 (1977) 177–188. Eckhard Platen: School of Finance and Economics and Department of Mathematical Sciences, University of Technology, Sydney, Broadway, NSW, 2007, Australia E-mail address: [email protected] Jason West: Department of Accounting, Finance and Economics, Griffith Business School, Griffith University, Nathan, QLD 4111, Australia E-mail address: [email protected]

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STOCHASTIC ANALYSIS OF BACKWARD TIDAL DYNAMICS EQUATION HONG YIN

Abstract. The backward stochastic tidal dynamics equations, a system of coupled backward stochastic differential equations, in bounded domains are studied in this paper. Under suitable projections and truncations, a priori estimates are obtained, which enable us to establish the uniformly boundedness of an adapted solution to the system. Such regularity does not usually hold for stochastic differential equations. The well-posedness of the projected system is given by means of the contraction property of the elevation component. The existence of solutions are proved by utilizing the Galerkin approximation scheme and the monotonicity properties for bounded terminal conditions. The uniqueness and continuity of solutions with respect to terminal conditions are also provided.

1. Introduction Tide, the alternate rising and falling of the sea levels, is the result of the combination of the rotation of the Earth, and the gravitational attraction exerted at different parts of the Earth by the Moon and the Sun. The study of ocean tides trace back to the early seventeenth century. The first attempt of a theoretical explanation of ocean tides was given by Galileo Galilei[6] in 1632. Although not very successful, it inspired many successors to further the studies in this field, including Johannes Kepler and Isaac Newton[25]. The latter established a scientific formulation in 1687, which pointed out the role of the lunar and solar gravitational effect on ocean tides. Later Maclaurin[20] used Newton’s theory of fluxion and took into account Earth’s rotational effects on motion, Euler discovered that the horizontal component of the tidal force, as opposed to the vertical component, is the main driving force of ocean tides that causes the wavelike progression of high tide, and Jean le Rond d’Alembert observed tidal equations for the atmosphere which did not include rotation. A major break through of the mathematical formulation of ocean tides is accredited to Laplace[14], who introduced a system of three linear partial differential equations for the horizontal components of ocean velocity, and the vertical displacement of the ocean surface in 1775. His work remains the basis of tidal computation to this day, and was followed up by Thomson and Tait[33], and Poincaré[28], among others. The former applied systematic harmonic analysis to tidal analysis and rewrote Laplace’s equations in terms of vorticity. In the first half of the nineteenth century, researchers further expanded the studies of ocean tides. Encouraging developments include Arctic tides, and geophysical tides such as tidal motion in the atmosphere and in the Earth’s molten core. In the second Received 2011-7-26; Communicated by P. Sundar. 2000 Mathematics Subject Classification. Primary 60H15, 35Q30; Secondary 76D05, 35R60. Key words and phrases. Backward stochastic tidal dynamics equation, backward stochastic differential equations, coupled backward stochastic differential equations, the Itô formula, contraction mapping, method of monotonicity, infinite dimensional analysis, Galerkin approximation. 745

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half of the nineteenth century, quantum advances in computers, satellite technology and numerical methods of solving PDEs made it possible to solve Laplace tidal equations with realistic boundary conditions and depth functions. In this article, we consider the tidal dynamics equations constructed by Marchuk and Kagan[21, 22]. The existence and uniqueness of the tide equations in forward case have been shown by Ipatova[9], Marchuk and Kagan[21], and Manna, Menaldi and Sritharan [23]. The backward version of stochastic tidal dynamics equations is, to our best knowledge, new. It appears as an inverse problem wherein the velocity profile and elevation component at a time T are observed and given, and the noise coefficient has to be ascertained from the given terminal data. Such a motivation arises naturally when one understands the importance of inverse problems in partial differential equations (see J. L. Lions [15, 16]). Since the problem of specifying the function of boundary condition on the liquid boundary, and the problem of specifying the tide-generating forces must be solved simultaneously with the tide theory equation system, Agoshkov[2], among other authors, has considered tidal dynamics models as inverse problems. Some studies of backward stochastic analysis on fluid dynamics has been put forth in our previous work [31]. Linear backward stochastic differential equations were introduced by Bismut in 1973 ([3]), and the systematic study of general backward stochastic differential equations (BSDEs for short) were put forward first by Pardoux and Peng[27], Ma, Protter, Yong, Zhou, and several other authors in a finite-dimensional setting. Ma and Yong[19] have studied linear degenerate backward stochastic differential equations motivated by stochastic control theory. Later, Hu, Ma and Yong [8] considered the semi-linear equations as well. Backward stochastic partial differential equations were shown to arise naturally in stochastic versions of the Black-Scholes formula by Ma, Protter and Yong [17, 18]. A nice introduction to backward stochastic differential equations is presented in the book by Yong and Zhou [34], with various applications. The usual method of proving existence and uniqueness of solutions by fixed point arguments do not apply to the stochastic system on hand since the drift coefficient in the backward stochastic tidal flow is nonlinear, non-Lipschitz and unbounded. However, the drift coefficient is monotone on bounded L4 (G) balls in H10 (G), which was first observed by Manna, Menaldi and Sritharan [23]. One may also refer to Menaldi and Sritharan [24] for more information. The Galerkin approximation scheme is employed in the proof of existence and uniqueness of solutions to the system. To this end, a priori estimates of finite-dimensional projected systems are studied, and uniformly boundedness of adapted solutions are established. Such regularity does not usually hold for stochastic differential equations. The well-posedness of the projected system is also given by means of the contraction property of the elevation component. In order to establish the monotonicity property of the drift term, a truncation of R(x), the depth of the calm sea, is introduced. Then the generalized Minty-Browder technique is used in this paper to prove the existence of solutions to the tidal dynamics system. The proof of the uniqueness and continuity of solutions are wrought by establishing the closeness of solutions of the system via monotonicity arguments. The structure of the paper is as follows. The functional setup of the paper is introduced and several frequently used inequalities are listed in section 2. Some a priori estimates for the solutions of the projected system are given under different assumptions on the terminal conditions and external force in section 3. Section 4 is devoted to well-posedness of the

BACKWARD STOCHASTIC TIDAL DYNAMICS EQUATION

747

projected system. The existence of solutions of the tidal dynamics equations under suitable assumptions is shown by Minty-Browder monotonicity argument in section 5. The uniqueness and continuity of the solution under the assumption that terminal condition is uniformly bounded in H 1 sense are given in section 6. 2. Formulation of the Problem Let us consider the time interval [0, T ], and let G, the horizontal ocean basin where tides are induced, be a bounded domain in R2 with smooth boundary conditions. The boundary contour ∂G is composed of two disconnected parts, a solid part of Γ1 coinciding with the edge of the continental and island shelves, and an open boundary Γ2 . Let us assume that sea water is incompressible and the vertical velocities are small compared with the horizontal velocities. Thus we are able to exclude acoustic waves. Also long waves, including tidal waves, are stood out from the family of gravitational oscillations. Furthermore, to reduce computational difficulties, we assume that the Earth is absolutely rigid, and the gravitational field of the Earth is not affected by movements of ocean tides. Also the effect of the atmospheric tides on the ocean tides and the effect of curvature of the surface of the Earth on horizontal turbulent friction are ignored. Under these commonly used assumptions, we are able to adopt the following tide dynamics model:  ∂w r    ∂t + lk × w = −g∇ζ − R |w|w + κh △w + g;    ∂ζ    ∂t + ∇ · (Rw) = 0; (2.1)    w = w0 on [0, T ] × ∂G;    R   w0 = 0 on Γ1 , and w0 dΓ2 = 0, Γ 2

where w, the horizontal transport vector, is the averaged integral of the velocity vector over the vertical axis, l = 2ρ cos θ is the Coriolis parameter, where ρ is the angular velocity of the Earth rotation and θ is the colatitude, k is an unit vector oriented vertically upward, g is the free fall acceleration, r is the bottom friction factor, κh is the horizontal turbulent viscosity coefficient, g is the external force vector, and ζ is the displacement of the free surface with respect to the ocean floor. The function w0 is an known function 0 on R the boundary. The restriction w |Γ1 = 0 is the no-slip condition on the shoreline, and 0 w dΓ2 = 0 follows from the mass conservation law. Here R is the vertical scale of moΓ2 tion, i.e., the depth of the calm sea. Let us assume that R is a continuously differentiable function of x, so that inf x∈G {R(x)} ≥ C0 and supx∈G {R(x) + |∇R(x)|} ≤ C1 for some positive constants C0 and C1 . In order to simplify the non-homogeneous boundary value problem to a homogeneous Dirichlet boundary value problem, we set u(t, x) = w(t, x) − w0 (t, x), and ξ(t, x) = ζ(t, x) +

Z

t

0

Let us denote by A the matrix A=

−κh △ 2ρ cos θ

∇ · R(x)w0 (s, x) ds. ! −2ρ cos θ , −κh △

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HONG YIN

and γ(x) ,

where

r R(x) .

Thus we are able to rewrite the tide dynamics model as  ∂u 0 0   on [0, T ] × G;  ∂t = −Au − γ|u + w |(u + w ) − g∇ξ + f    ∂ξ    ∂t + ∇ · (Ru) = 0;    u = 0 on [0, T ] × ∂G;      u = u0 and ξ = ξ0 at t = 0,

(2.2)

 Rt 0   + g∇ 0 ∇ · (Rw0 )ds + κh △w0 − lk × w0 ; f = g − ∂w  ∂t    u0 (x) = w0 (x) − w0 (0, x);      ξ0 (x) = ζ0 (x).

To unify the language, let us introduce the following definitions and notations. Definition 2.1. Let A be an operator on a separable Hilbert space K with complete orthonormal system (CONS for short) {e j }∞j=1 . If hAx, yi=hx, A∗ yi for any x, y ∈ K, then A∗ is called the adjoint of A. If A = A∗ , then A is called self-adjoint. Definition 2.2. Let A be a linear operator from a separable Hilbert space K with CONS {e j }∞j=1 to a separable Hilbert space H.

(a) We denote by L(K, H) the class of all bounded linear operators with the uniform operator norm k · kL . P 1 ∗ 2 (b) If kAkL1 = ∞ k=1 h(A A) ek , ek iK < ∞, then A is called a trace class(nuclear) operator. We denote by L1 (K, H) the class of trace class operators equipped with norm k · kL1 . (c) We also denote by L2 (K, H) the class of Hilbert-Schmidt operators with norm P 1 2 k · kL2 given by kAkL2 =( ∞ k=1 hAek , Aek iH ) . Sometimes k · kL2 is also denoted by k · kH.S . (d) Let Q ∈ L1 (K, K) be self-adjoint and positive definite. Let K0 be the Hilbert subspace of K with inner product 1

1

h f, giK0 = hQ− 2 f, Q− 2 giK , and we denote LQ = L2 (K0 , H) with the inner product hF, GiLQ = tr(FQG∗ ) = tr(GQF ∗ ),

F, G ∈ LQ .

Definition 2.3. A stochastic process W(t) is called an H-valued Q-Wiener process, where Q is a trace class operator on H, if W(t) satisfies the following: (a) W(t) has continuous sample paths in H-norm with W(0) = 0. (b) (W(t), h) has stationary independent increments for all h ∈ H. (c) W(t) is a Gaussian process with mean zero and covariance operator Q, i.e. E(W(t), g)(W(s), h) = (t ∧ s)(Qg, h) for all g, h ∈ H. Let L2 (G) and H10 (G) be standard Sobolev spaces with norms Z kuk2L2 , |u|2 dx G


749

and kuk2H1 , 0

Z

G

|∇u|2 dx,

respectively. Denote H−1 (G) the dual space of H10 (G). Denote (·, ·) the inner product of L2 (G), (·, ·)H10 the inner product of H10 (G), and h·, ·i the duality paring between H10 (G) and H−1 (G). Let k · kL2 be the norm of L2 and k · kH10 be the norm of H10 (G). Similarly, we can define the norms, inner products of L2 (G), H01 (G) and H −1 (G). It is clear that H10 (G) ⊂ L2 (G) ⊂ H−1 (G) and

H01 (G) ⊂ L2 (G) ⊂ H −1 (G)

are Gelfand triples, and for any x ∈ L2 (G) and y ∈ H10 (G), there exists x′ ∈ H−1 (G), such that (x, y) = hx′ , yi. The mapping x 7→ x′ is linear, injective, compact and continuous. A similar result holds for L2 (G), H01 (G) and H −1 (G). Remark 2.4. (i) Let Q be a trace class operator on L2 (G). Let {e j }∞j=1 ∈ L2 (G) ∩ H10 (G) ∩ L4 (G) be a CONS in L2 (G) such that there exists a nondecreasing sequence of positive P numbers {λ j }∞j=1 , lim j→∞ λ j = ∞ and −△e j = λ j e j for all j. Let Qek = qk ek with ∞ k=1 qk < k 2 ∞, and {b (t)} be a sequence of iid Brownian motions in R. Then the L (G)-valued QP √ k Wiener process is taken as W(t)= ∞ qk b (t)ek . k=1 (ii) Let Q be a trace class operator on L2 (G). Similarly, we can define a complete orthonormal system {e j }∞j=1 , a nondecreasing sequence of positive numbers {ρ j }∞j=1 such P that −△e j = ρ j e j , and positive numbers q′j such that Qe j = q′j e j and ∞j=1 q′j < ∞. Let P q W(t)= ∞j=1 q′j b j (t)e j . Then W(t) is an L2 (G)-valued Q-Wiener process.

Thus according to Definition 2.2 and 2.3, LQ , the space of linear operators E such that 1 EQ 2 is a Hilbert-Schmidt operator from L2 (G) to L2 (G), is well-defined, and so is LQ . In this paper we consider a filtered complete probability space (Ω, F , P; {Ft }t≥0 ), where {Ft } is the natural filtration of {W(t)} and {W(t)}, augmented by all the P-null sets of F . Introducing randomness to system (2.2), and suppose the terminal value of the tide is given, one can construct the following backward stochastic tidal dynamics equations:  ∂u(t) dW(t) 0 0    ∂t = −Au(t) − γ|u(t) + w (t)|(u(t) + w (t)) − g∇ξ(t) + f(t) + Z(t) dt ;    ∂ξ(t) dW(t) (2.3)   ∂t + ∇ · (Ru(t)) = Z(t) dt ;    u(T ) = φ and ξ(T ) = ψ,

where f ∈ L2 (0, T ; H−1 ), φ ∈ L2FT (Ω; L2 (G)) and ψ ∈ L2FT (Ω; L2 (G)).

Definition 2.5. A quaternion of Ft -Adapted processes (u, Z, ξ, Z) is called a solution of backward tidal dynamics equation (2.3) if it satisfies the integral form of the system o RTn    u(t) = φ + t Au(s) + γ|u(s) + w0 (s)|(u(s) + w0 (s)) + g∇ξ(s) − f(s) ds    RT  − t Z(s)dW(s);    R R   ξ(t) = ψ + T ∇ · (Ru(s))ds − T Z(s)dW(s); t

t

P-a.s., and the following holds: (a) u ∈ L2F (Ω; L∞ (0, T ; L2(G))) ∩ L2F (Ω; L2 (0, T ; H10 (G))); (b) Z ∈ L2F (Ω; L2 (0, T ; LQ ));

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HONG YIN

(c) ξ ∈ L2F (Ω; L∞ (0, T ; L2 (G))) ∩ L2F (Ω; L2 (0, T ; H01(G))); (d) Z ∈ L2F (Ω; L2 (0, T ; LQ )). We list below some commonly used results and omit some of the proofs. Readers may refer to Adams[1], Kesavan[11], Ladyzhenskaya[12], Manna, Menaldi and Sritharan[23], and Temam[32] for more details. Lemma 2.6. For any real-valued, compact supported smooth functions x and y in R2 , the following holds: kxyk2L2 ≤ kx∂1 xkL1 ky∂2 ykL1 , kxk4L4 ≤ 2kxk2L2 k∇xk2L2 . Lemma 2.7. Let X be a normed linear space. Let O be an open subset of X, and K be a convex subset of O. Let J : O → R be twice differentiable in O. Then J is convex if and only if, for all u and v ∈ K, J ′′ (v; u, u) =

d2 J(v + θu + αu)|θ,α=0 ≥ 0. dθdα

Lemma 2.8. Denote B(u) , γ|u + w0 |(u + w0 ). Then B(·) is a continuous operator from L4 (G) into L2 (G), and for all u and v in L4 (G), hB(u) − B(v), u − vi ≥ 0. Lemma 2.9.

(a) For any u and v ∈ H10 (G), and u has a smooth second derivative, (Au, u) = κh kuk2H1 0

and (Au, v) ≤ C2 kukH10 kvkH10 for some constant C2 = κh + 2ρ cos θ. (b) For any u and w0 ∈ L4 (G), kB(u)kL2 ≤ C3 kukL4 , where C3 = sup x∈G γ(x). (c) For any u, v and w0 ∈ L4 (G),

and

n o kB(u) − B(v)kL2 ≤ C3 kukL4 + kvkL4 ku − vkL4 , n o |hB(u) − B(v), u − vi| ≤ C4 kuk2L4 + kvk2L4 ku − vkL2 .

(d) For any u, v ∈ H10 (G) and w0 ∈ L4 (G),

n o 3 1 |hB(u) − B(v), u − vi| ≤ C5 kukL4 + kvkL4 ku − vkL2 2 ku − vkH2 1 . 0


751

3. A Priori Estimates In this section we are going to show some a priori estimates for a projected system. These projections are useful for the Galerkin approximation scheme employed in Section 5 and Section 6. For any N ∈ N, let L2 N (G) , span{e1 , e2 , · · · , eN }

be the N-dimensional subspace of L2 (G). Likewise, we can define H10N (G), H−1 N (G), 1 L2N (G), H0N (G) and HN−1 (G). Note that since {e j }∞j=1 ∈ L2 (G) ∩ H10 (G) ∩ L4 (G), we have L2 N (G) = H10 N (G) = H−1 N (G). Similarly, we have 1 L2N (G) = H0N (G) = HN−1 (G).

Let PN be the orthogonal projection from L2 (G) to L2 N (G). Let WN (t) , PN W(t) and W N (t) , PN W(t). PN p ′ i PN √ i qi b (t)ei and W N (t) = i=1 qi b (t)ei . Let Note that by Remark 2.4, WN (t) = i=1 N N N {Ft } be the natural filtration of {W (t)} and {W (t)}, and we introduce the following projections: f N (t) , PN f(t), φN , E(PN φ|FTN ) and ψN , E(PN ψ|FTN ). The projected backward tide dynamics system is given by  ∂uN (t)   = −AuN (t) − BN (uN (t)) − g∇ξ N (t) + f N (t) + ZN (t)dWN (t);     ∂ξ∂tN (t) N N N N   ∂t + ∇ · (R u (t)) = Z (t)dW (t);    uN (T ) = φN and ξ N (T ) = ψN ,

(3.1)

where BN (u) , γN |u + w0N |(u + w0N ) for all u ∈ L4 (G).

2 Proposition 3.1. Suppose that the terminal conditions satisfy φ ∈ L∞ FT (Ω; L (G)), ψ ∈ ∞ 2 2 −1 LFT (Ω; L (G)), and the external force f ∈ L (0, T ; H (G)). Then for any solution of system (3.1), the following is true: n o 2 2 2 2 2 1 (uN , ZN ) ∈ L∞ F ([0, T ] × Ω; L (G)) ∩ LF (Ω; L (0, T ; H0 (G))) × LF (Ω; L (0, T ; LQ )), n o 2 2 2 1 2 2 (ξ N , Z N ) ∈ L∞ F ([0, T ] × Ω; L (G)) ∩ LF (Ω; L (0, T ; H0 (G))) × LF (Ω; L (0, T ; LQ )).

Proof. An application of the Itô formula to kuN (t)k2L2 yields Z T kuN (t)k2L2 + kZN (s)k2LQ ds t Z T N 2 =kφ kL2 + 2 hAuN (s) + BN (uN (s)) + g∇ξ N (s) − f N (s), uN (s)ids t Z T N −2 hZ (s)dWN (s), uN (s)i.

(3.2)

t

By Lemma 2.6 and 2.9, we have

2hAuN (s), uN (s)i = 2(AuN (s), uN (s)) = 2κh kuN (s)k2H1 , 0

(3.3)

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HONG YIN

2hBN (uN (s)), uN (s)i ≤2C3 kuN (s)kL4 kuN (s)kL2 3

1

≤4C3 kuN (s)kL2 2 kuN (s)kH2 1 0

≤3C3 kuN (s)k2L2 + C3 kuN (s)k2H1 ,

(3.4)

0

2hg∇ξ N (s), uN (s)i = − 2ghξ N (s), ∇ · uN (s)i

≤2gkξ N (s)kL2 k∇ · uN (s)kL2 ≤gkξ N (s)k2L2 + gkuN (s)k2H1

(3.5)

0

and 2h−f N (s), uN (s)i ≤ kf N (s)k2H−1 + kuN (s)k2H1 .

(3.6)

0

Thus (3.2) becomes Z T N 2 ku (t)kL2 + kZN (s)k2LQ ds t Z T N 2 ≤kφ kL2 + (2κh + C3 + g + 1)kuN (s)k2H1 + 3C3 kuN (s)k2L2 + gkξ N (s)k2L2 0 t Z T hZN (s)dWN (s), uN (s)i. + kf N (s)k2H−1 ds − 2

(3.7)

t

Applying the Itô formula to kξ N (s)k2L2 to get Z T N 2 kξ (t)kL2 + kZ N (s)k2LQ ds t Z T Z N 2 N N N =kψ kL2 + 2h∇ · (R u (s)), ξ (s)ids − t

T

2hZ N (s)dW N (s), ξ N (s)i.

(3.8)

t

The term

2h∇ · (RN uN (s)), ξ N (s)i

=2hRN ∇ · uN (s), ξ N (s)i + 2huN (s)∇RN , ξ N (s)i

≤2kRN kL∞ kuN (s)kH10 kξ N (s)kL2 + 2kuN (s)kL2 k∇RN kL2 kξ N (s)kL2 n o ≤C1 kuN (s)k2L2 + kuN (s)k2H1 + 2kξ N (s)k2L2 .

(3.9)

0

Thus substituting (3.9) into (3.8), and adding up (3.7) and (3.8), one gets Z T Z T E Fr kuN (t)k2L2 + E Fr kξ N (t)k2L2 + E Fr kZN (s)k2LQ ds + E Fr kZ N (s)k2LQ ds t t Z T Fr N 2 Fr N 2 Fr ≤E kφ kL2 + E kψ kL2 + E (2κh + C3 + g + 1 + C1 )kuN (s)k2H1 0 t + (3C3 + C1 )kuN (s)k2L2 + (g + 2)kξ N (s)k2L2 + kf N (s)k2H−1 ds (3.10)


for 0 ≤ r ≤ t, P-a.s. Since

753

kuN (s)k2H1 0

=h−△uN (s), uN (s)i =

N X i=1

hλi ei , uN (s)i

≤λN kuN (s)k2L2 , where λi , as stated in Section 2, is the eigenvalue of −△ with respect to ei . Thus equation (3.10) becomes Z Tn o E Fr kuN (t)k2L2 + E Fr kξ N (t)k2L2 + E Fr kuN (s)k2H1 + kξ N (s)k2H 1 ds 0 0 t Z T Z T + E Fr kZN (s)k2LQ ds + E Fr kZ N (s)k2LQ ds (3.11) t t Z T ≤E Fr kφN k2L2 + E Fr kψN k2L2 + E Fr K(N)kuN (s)k2L2 + K(N)kξ N (s)k2L2 + kf N (s)k2H−1 ds, t

P-a.s., where K(N) is a constant depending on N only. By means of the Gronwall inequality and letting r = t, one obtains Z Tn n o o sup kuN (t)k2L2 + kξ N (t)k2L2 + E kuN (s)k2H1 + kξ N (s)k2H 1 ds t∈[0,T ]

+E

0

Z

T

0

kZN (s)k2LQ ds + E

Z

0

0

kZ N (s)k2LQ ds

Z n ≤K(N) sup E Ft kφN k2L2 + sup E Ft kψN k2L2 + t∈[0,T ]

P-a.s., which completes the proof.

0

T

t∈[0,T ]

0

T

o kf N (s)k2H−1 ds ,

(3.12)

Proposition 3.2. Suppose that the terminal conditions satisfy φ ∈ LnFT (Ω; L2 (G)), ψ ∈ LnFT (Ω; L2 (G)), and the external force f ∈ L2 (0, T ; H−1 (G)), for all n ∈ N and n ≥ 2. The following is true for any solution of system (3.1): n o (uN , ZN ) ∈ L∞ (0, T ; LnF (Ω; L2 (G))) ∩ LnF (Ω; Ln (0, T ; H10 (G))) × L2F (Ω; L2 (0, T ; LQ)), n o (ξ N , Z N ) ∈ L∞ (0, T ; LnF (Ω; L2 (G))) ∩ LnF (Ω; Ln (0, T ; H01(G))) × L2F (Ω; L2 (0, T ; LQ )).

Proof. First of all, the case when n = 2 can be proved by applying the Gronwall inequality to equation (3.11) in Proposition 3.1, taking the expectation, and then taking supremum over the time interval [0, T ]. Secondly, suppose the proposition holds for all 2 ≤ m ≤ n−1. Let us show that the proposition is still true for m = n. An application of the Itô formula yields Z n2 − n T N N 2 ku (s)kn−2 kuN (t)knL2 + L2 kZ (s)kLQ ds 2 t Z T N n N N N N N N N =kφ kL2 + n ku (s)kn−2 L2 hAu (s) + B (u (s)) + g∇ξ (s) − f (s), u (s)ids t

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HONG YIN

−n

Z

t

T N N N kuN (s)kn−2 L2 hZ (s)dW (s), u (s)i.

Taking the expectation on both sides, one obtains Z T Z T n2 − n N 2 E kuN (s)kn−2 EkuN (t)knL2 + E kuN (s)knH1 ds + L2 kZ (s)kLQ ds 0 2 t t Z T =EkφN knL2 + nE kuN (s)kn−2 hAuN (s) + BN (uN (s)) + g∇ξ N (s)ids L2 t Z T Z T N N 1 kf (s)kH−1 ds + E − nE kuN (s)kn−2 ku (s)k kuN (s)knH1 ds 2 H0 L 0 t t Z T Z T ≤EkφN knL2 + K(n, N)E kuN (s)knL2 ds + K(n, N)E kξ N (s)knL2 ds t t p Z T N kf (s)kH−1 EkuN (s)kn−1 ds + n λN L2 t Z T Z T ≤EkφN knL2 + K(n, N)E kuN (s)knL2 ds + K(n, N)E kξ N (s)knL2 ds t t Z T p + n λN sup EkuN (s)kn−1 kf N (s)kH−1 ds. L2 s∈[0,T ]

(3.13)

t

where K(n, N) is a constant depending on n and N only. Similar, one can show that Z T Z T n2 − n Ekξ N (t)knL2 + E kξ N (s)knH 1 ds + E kξ N (s)kn−2 kZ N (s)k2LQ ds L2 0 2 t t Z Tn o N n N n N ≤Ekψ kL2 + K(n, N)E ku (s)kL2 + kξ (s)knL2 ds. (3.14) t

Adding up (3.13) and (3.14) to get

Z T kuN (s)knH1 ds + E kξ N (s)knH 1 ds 0 0 t t Z Tn o ≤EkφN knL2 + EkψN knL2 + K(n, N)E kuN (s)knL2 + kξ N (s)knL2 ds t Z T + K(n, N) kf N (s)knH−1 ds,

EkuN (t)knL2 + Ekξ N (t)knL2 + E

Z

T

t

which completes the proof after an application of the Gronwall inequality.

4. Well-posedness of the Projected System In this section, we are going to show the well-posedness of the projected system (3.1). In order to do so, we need to truncate the system. For every M ∈ N, let L M to be the Lipschitz C ∞ function given as follows:    1 if kukH10 < M     L M (kukH10 ) =  0 if kukH10 > M + 1     0 ≤ L M (kukH1 ) ≤ 1 otherwise 0


755

Proposition 4.1. kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)kL2 ≤ C(N, M)kx − ykH10 for any x, y ∈ L2 N (G) and M ∈ N, where C(N, M) is a constant depending on N, M and G only. Proof. Let x and y be any two elements in L2 N (G). Without lose of generality, we assume that kxkH10 ≤ kykH10 , and let us discuss it in the following 3 cases: Case I. kxkH10 > M + 1. By the definition of L M , kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)k2L2 = 0 ≤ kx − yk2H1 . Thus 0

we see that L M BN is Lipschitz. Case II. kykH10 ≤ M + 1. It is clear that

kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)k2L2 =

N X i=1

=

N X i=1

≤2 ≤2

|hL M (kxkH10 )BN (x) − L M (kykH10 )BN (y), ei i|2 |hL M (kykH10 )BN (x) − L M (kykH10 )BN (y) + L M (kxkH10 )BN (x) − L M (kykH10 )BN (x), ei i|2

N X i=1

N X i=1

|L M (kykH10 )hBN (x) − BN (y), ei i|2 + 2

N X i=1

L2M (kykH10 )|hBN (x) − BN (y), ei i|2 + 2C 2M

|hBN (x), ei i|2 |L M (kxkH10 ) − L M (kykH10 )|2 N X i=1

|hBN (x), eii|2 kx − yk2H1 ,

(4.1)

0

where C M is Lipschitz coefficient of L M . By Lemma 2.6, Lemma 2.9, and the Poincaré inequality, one has N X i=1

=

|hBN (x) − BN (y), ei i|2

N X i=1

kBN (x) − BN (y)k2L2

h i ≤2C32 kxk2L4 + kyk2L4 kx − yk2L4 h i ≤4C32CG kxk2H1 + kyk2H1 kx − yk2H1 , 0

0

0

where CG is a constant depending on G only. Also N X i=1

Thus (4.1) becomes

|hBN (x), ei i|2 ≤ C22 kxk2L4 ≤ C22CG kxk2H1 . 0

kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)k2L2 o i n h ≤ 8C32CG L2M (kykH10 ) kxk2H1 + kyk2H1 + 2C 2M C22CG kxk2H1 kx − yk2H1 . 0

0

0

0

Since kxkH10 and kykH10 are all bounded by M + 1, kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)k2L2 ≤ C(N, M)kx − yk2H1 , where C(N, M) is only related to N, M, and G. 0

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HONG YIN

Case III. kykH10 > M + 1 and kxkH10 ≤ M + 1. Then by the definition of L M , L M (kykH10 ) = 0. Thus kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)k2L2 ≤ 2C 2M C22CG kxk2H1 kx − yk2H1 . 0

0

Thus we have shown that kL M (kxkH10 )BN (x) − L M (kykH10 )BN (y)kL2 ≤ C(N, M)kx − ykH10 ,

where C(N, M) is a constant which is only related to N, M and G. Let us state without proof an useful result from Yong and Zhou [34].

Proposition 4.2. For any (y, z) ∈ Rk ×Rk×m , assume that h(t, y, z) : [0, T ]×Rk ×Rk×m ×Ω → Rk is {Ft }t≥0 -adapted with h(·, 0, 0)∈ L2F (Ω; L2 (0, T ; Rk )). Moreover, there exists an L > 0, such that |h(t, y, z) − h(t, y¯ , z¯)| ≤ L{|y − y¯ | + |z − z¯|} k ∀t ∈ [0, T ], y, y¯ ∈ R and z, z¯ ∈ Rk×m P-a.s. For any given ξ∈L2FT (Ω; Rk ), the BSDE    dY(t) = h(t, Y(t), Z(t))dt + Z(t)dW(t), t ∈ [0, T ), a.s. (4.2)   Y(T ) = ξ, admits a unique adapted solution (Y(·), Z(·))∈M[0, T ], where

M[0, T ] = L2F (Ω; C([0, T ]; R)) × L2F (Ω; L2 (0, T ; R))

and it is equipped with the norm

kY(·), Z(·)kM[0,T ] = {E( sup |Y(t)|2 ) + E 0≤t≤T

Z

T

0

1

|Z(t)|2 dt} 2 .

Now we are able to prove the main result of this section. Theorem 4.3. System (3.1) admits a unique adapted solution (uN , ZN , ξ N , Z N ) in n o 2 2 2 1 2 2 L∞ F ([0, T ] × Ω; L (G)) ∩ LF (Ω; L (0, T ; H0 (G))) × LF (Ω; L (0, T ; LQ )) n o 2 2 2 1 2 2 × L∞ F ([0, T ] × Ω; L (G)) ∩ LF (Ω; L (0, T ; H0 (G))) × LF (Ω; L (0, T ; LQ )),

provided that the terminal conditions satisfy φ ∈ L∞ (Ω; L2 (G)), ψ ∈ L∞ (Ω; L2 (G)), and FT FT 2 −1 the external force f ∈ L (0, T ; H (G)). Proof. First of all, for any M ∈ R, let us define a truncated system as follows:  ∂uN M (t)   = −AuN M (t) − L M (kuN M (t)kH10 )BN (uN M (t)) − g∇ξ N M (t) + f N (t)  ∂t      +ZN M (t)dWN (t);  NM   ∂ξ (t)   + ∇ · (RN uN M (t)) = Z N M (t)dW N (t);  ∂t    uN M (T ) = φN and ξ N M (T ) = ψN .

(4.3)

From Lemma 4.1, it is clear that all coefficients of the above system are Lipschitz continuous for fixed N and M. Let us fix ζ(t) ∈ L∞ ([0, T ] × Ω; L2N (G)) ∩ L2F (Ω; L2 (0, T ; H01 N (G))). F Consider  NM   ∂u ∂t (t) = −AuN M (t) − L M (kuN M (t)kH1 )BN (uN M (t)) − g∇ζ(t) + f N (t) + ZN M (t)dWN (t);  0   uN M (T ) = φN . (4.4)


757

Let us map system (4.4) to RN . It is obviously that the image of system is equivalent to system (4.4). Since the coefficients in the image system are Lipschitz, Proposition 4.2 guarantees the existence of a unique adapted solution of the image system. By the equivalence between two systems, we claim that system (4.4) admits a unique adapted solution (uN M , ZN M ). Clearly, for this uN M , the following system  NM ∂ξ (t)    ∂t + ∇ · (RN uN M (t)) = Z N M (t)dW N (t); (4.5)   ξ N M (T ) = ψN

admits a unique adapted solution (ξ N M , Z N M ). Hence we can define an operator Φ, such that Φ(ζ) , ξ N M . We would like to show that Φ is a contraction mapping. For any ζ1 and 2 2 2 1 NM ζ2 ∈ L∞ and Φ(ζ2 ) = ξ2N M . F ([0, T ] × Ω; LN (G)) ∩ LF (Ω; L (0, T ; H0 N (G))), let Φ(ζ1 ) = ξ1 Denote uˆ , u1N M − u2N M , ξˆ , ξ1N M − ξ2N M , ζˆ , ζ1 − ζ2 , ˆ , ZN M − ZN M , Zˆ , Z N M − Z N M . Z 1 2 1 2 Similar to the proof of Proposition 4.1, one can verify that ˆ |hL M (ku1N M (t)kH10 )BN (u1N M (t)) − L M (ku2N M (t)kH10 )BN (u2N M (t)), u(t)i| ˆ 2H1 , ≤C(M, CG , C3 )ku(t)k 0

where C(M, CG , C3 ) is a constant depending on M, CG , C3 only. Let η be a positive number such that ρN 2κh λN + C(M, CG , C3 )λN + λN η > max( , ). 2 2 ˆ 2 2 e2ηt to get Applying the Itô formula to kξ(t)k L Z T 2 2ηs ˆ 2 2 e2ηt + ˆ kξ(t)k kZ(s)k LQ e ds L t Z Tn Z T o 2 N 2ηs N ˆ ˆ ˆ ˆ ˆ = − 2ηkξ(s)kL2 + 2h∇ · (R u(s)), ξ(s)i e ds − 2e2ηs hZ(s)dW (s), ξ(s)i t t Z Tn o 2 2ηs ˆ ˆ ˆ 1 kξ(s)k 2 e ds ≤ − 2ηkξ(s)k + 2C (1 + C )k u(s)k 2 1 G L H L 0 t Z T N ˆ − 2e2ηs hZ(s)dW (s), ξˆ(s)i, t

where the estimates are obtained similar to (3.9). Thus for any 0 ≤ r ≤ t, Z T Z T 2 2ηs Fr 2 2ηs ˆ ˆ 2 2 e2ηt + E Fr ˆ E Fr kξ(t)k k ξ(s)k e ds + E kZ(s)k ds 1 LQ e L H0 t t Z Tn o 2 2 2ηs ˆ ˆ ˆ ˆ ≤E Fr − 2ηkξ(s)k ds 2 + kξ(s)k 1 + 2C 1 (1 + CG )ku(s)k H1 kξ(s)kL2 e L

t

≤E Fr

≤E Fr

Z

T

t

Z

t

T

n

H0

0

o 2 2ηs ˆ ˆ ˆ (−2η + ρN )kξ(s)k H10 kξ(s)kL2 e ds L2 + 2C1 (1 + CG )ku(s)k

e2ηs

C12 (1 + CG )2 2 ˆ ku(s)k ds, H10 2η − ρN

(4.6)

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HONG YIN

ˆ 2L2 e2ηt P-a.s., where ρN is defined in Remark 2.4. An application of the Itô formula to ku(t)k yields Z T 2 2ηs ˆ ˆ 2L2 e2ηt + ku(t)k kZ(s)k LQ e ds t Z Tn 2 NM N NM ˆ ˆ 1 = − 2ηku(s)k L2 + 2hAu(s) + L M (ku1 (s)kH0 )B (u1 (s)) t o ˆ ˆ − L M (ku2N M (s)kH10 )BN (u2N M (s)) + g∇ζ(s), u(s)i e2ηs ds Z T N 2ηs ˆ ˆ − 2hZ(s)dW (s), u(s)ie t Z T n o p 2 2ηs ˆ 2 ˆ 2 ˆ ≤ − 2η + 2κh λN + C(M, CG , C3 )λN ku(s)k L2 + 2g λN kζ(s)kL ku(s)kL e ds t Z T N 2ηs ˆ ˆ − 2hZ(s)dW (s), u(s)ie . t

Thus for any 0 ≤ r ≤ t,

Z T Z T 2 2ηs 2 2ηs Fr ˆ ˆ 2L2 e2ηt + E Fr ˆ E Fr ku(t)k ku(s)k e ds + E kZ(s)k ds LQ e H10 t t Z T n 2 ˆ ≤E Fr − 2η + 2κh λN + C(M, CG , C3 )λN + λN ku(s)k L2 t o p 2ηs ˆ ˆ + 2g λN kζ(s)k L2 ku(s)k L2 e ds Z T g2 λ N 2 ˆ ≤E Fr e2ηs kζ(s)k L2 ds, 2η − 2κh λN − C(M, CG , C3 )λN − λN t

P-a.s. Equations (4.6) and (4.7) imply Z T Z Fr ˆ 2 2ηt Fr 2 2ηs Fr ˆ E kξ(t)kL2 e + E kξ(s)kH 1 e ds + E t

≤

0

(4.7)

T t

2 2ηs ˆ kZ(s)k ds LQ e

C12 (1 + CG )2 g2 λ N E Fr 2η − ρN 2η − 2κh λN − C(M, CG , C3 )λN − λN

Z

t

T 2 ˆ e2ηs kζ(s)k L2 ds,

P-a.s. Hence we take η to be large enough such that Z T Z 1 Fr T 2ηs ˆ 2 2ηs ˆ E Fr kξ(s)k e ds ≤ E e kζ(s)k2H 1 ds, H01 0 2 t t

P-a.s. Taking the expectation and letting t = 0, we see that Φ is a contraction mapping from L2F (Ω; L2 (0, T ; H01 N (G))) to L2F (Ω; L2 (0, T ; H01 N (G))). By the contraction mapping theorem, a unique adapted solution (uN M , ZN M , ξ N M , Z N M ) of (4.3) is guaranteed. As shown in Proposition 3.1, supt∈[0,T ] kuN (t)k2L2 ≤ K(N), where K(N) is a constant associated with N only. Since for finite-dimensional spaces, the norms k·kL2 and k·kH10 are equivalent, we know that kuN (s)kH10 is also uniformly bounded for every N. By the definition of the truncation L M , it is clear that (3.1) and (4.3) are equivalent when M is large enough. Thus letting M approach infinity, the limit of the solution (uN M , ZN M , ξ N M , Z N M ) is the unique adapted solution of the projected system (3.1). The regularity of the solution can be obtained by Proposition 3.1.


759

Continuity of the solution to the projected system (3.1) can also be obtained along similar lines of the proof of Theorem 6.1. We shall skip the proof and postpone it to Section 6. Thus the well-posedness of the projected system has been fully investigated. 5. Existence In this section, we are going to show the existence of an adapted solution of system (2.3). The Galerkin approximation scheme and Minty-Browder technique will be employed. In order to assure an uniform bound on a priori estimates, we make the following assumptions. Such an approach is commonly taken in the study of stochastic Euler equations by several authors so that a dissipative effect arises. Also they are standard hypotheses in the theory of stochastic PDEs in infinite dimensional spaces (see Chow [5], Kallianpur and Xiong [10], Prévôt and Röckner [29]). (A.1) (Continuity): f: H10 → H−1 is a continuous operator; (A.2) (Coercivity): There exist positive constants α and β, such that hAu − f(u), ui ≤ αkuk2L2 − βkuk2H1 ; 0

hAu − f(u), Aui ≤

αkuk2H1 0

− βkAuk2H1 ; 0

(A.3) (Monotonicity): There exist ν > 1 and α > 0, such that for any u and v in H10 , and M ∈ N, hνA(u − v) − (f(u) − f(v)), R M (u − v)i ≤ αku − vk2L2 ,

where R M is the projection of R into L2 M (G); (A.4) (Linear growth): For any u ∈ H10 and some positive constant α, |hf(u), ui| ≤ αkuk2 .

The system (2.3) can now be written as  ∂u(t) dW(t) 0 0    ∂t = −Au(t) − γ|u(t) + w (t)|(u(t) + w (t)) − g∇ξ(t) + f(u(t)) + Z(t) dt ;    ∂ξ(t) dW(t)   ∂t + ∇ · (Ru(t)) = Z(t) dt ;    u(T ) = φ and ξ(T ) = ψ,

and the corresponding projected system is  ∂uN (t)   = −AuN (t) − BN (uN (t)) − g∇ξ N (t) + f N (uN (t)) + ZN (t)dWN (t);     ∂ξ∂tN (t) N N N N   ∂t + ∇ · (R u (t)) = Z (t)dW (t);    uN (T ) = φN and ξ N (T ) = ψN .

(5.1)

(5.2)

Under these assumptions, we are able to prove a very important monotonicity result, which is the essence of proof of the existence theorem. 4

Lemma 5.1. For any u, v ∈ L 3 ([0, T ]; L4(G)) ∩ L0 (0, T ; H10 (G)), and M ∈ N, define Z T o 34 3 C14C54 13 n r(t) , 2α + 5 ku(s)kL4 + kv(s)kL4 ds. t 2 3 (ν − 1)κhC0 Then 1 hA(u − v) + B(u) − B(v) − (f(u) − f(v)) + r˙(t)(u − v), R M (u − v)i ≤ 0. 2

760

HONG YIN

Proof. From the monotonicity assumption and Lemma 2.9, one has hA(u − v) + B(u) − B(v) − (f(u) − f(v)), R M (u − v)i

=hνA(u − v) − (f(u) − f(v)), R M (u − v)i + hB(u) − B(v), R M (u − v)i

+ h(1 − ν)A(u − v), R M (u − v)i n o 3 1 ≤αku − vk2L2 + C1 C5 kukL4 + kvkL4 ku − vkL2 2 ku − vkH2 1 + (1 − ν)κh C0 ku − vk2H1 0

0

≤αku − vk2L2 +

3 8

23

o 43 C14C54 13 n kukL4 + kvkL4 ku − vk2L2 (ν − 1)κhC0

1 = − r˙(t)ku − vk2L2 , 2 which completes the proof.

The coercivity assumption assures a uniform a priori estimate. The following a priori estimate is very useful for the Galerkin approximation which will be used in Theorem 5.3. 2 Proposition 5.2. (i) Suppose that the terminal conditions satisfy φ ∈ L∞ FT (Ω; L (G)) and ∞ 2 ψ ∈ LFT (Ω; L (G)). Then for any solution of system (5.2), the following is true: n o 2 2 2 1 2 2 (uN , ZN ) ∈ L∞ F ([0, T ] × Ω; L (G)) ∩ LF (Ω; L (0, T ; H0 (G))) × LF (Ω; L (0, T ; LQ )), 2 2 2 (ξ N , Z N ) ∈ L∞ F ([0, T ] × Ω; L (G)) × LF (Ω; L (0, T ; LQ )).

Moreover,

sup kuN (t)k2L2 + E

t∈[0,T ]

+E

Z

T 0

Z

T

kuN (s)k2H1 ds + sup kξ N (t)k2L2 0

0

kZN (s)k2LQ ds + E

Z

0

t∈[0,T ]

T

kZ N (s)k2LQ ds ≤ K,

(5.3)

P-a.s. for some constant K, independent of N. (ii) Suppose the terminal conditions satisfy φ ∈ LnFT (Ω; L2 (G)) and ψ ∈ LnFT (Ω; L2 (G)) for all n ∈ N and n ≥ 2. The following is true for any solution of system (5.2): o n (uN , ZN ) ∈ L∞ (0, T ; LnF (Ω; L2 (G))) ∩ L2F (Ω; L2 (0, T ; H10(G))) × L2F (Ω; L2 (0, T ; LQ )), (ξ N , Z N ) ∈ L∞ (0, T ; LnF (Ω; L2 (G))) × L2F (Ω; L2 (0, T ; LQ )).

Moreover,

sup Eku

N

t∈[0,T ]

+E

Z

0

T

(t)knL2

+E

Z

0

T N 2 N n kuN (s)kn−2 L2 ku (s)kH1 ds + sup Ekξ (t)kL2

kZN (s)knLQ ds + E

0

Z

0

t∈[0,T ]

T

kZ N (s)knLQ ds ≤ K,

(5.4)

for some constant K, independent of N. 1 ∞ 1 (iii) Let the terminal conditions satisfy φ ∈ L∞ FT (Ω; H0 (G)) and ψ ∈ LFT (Ω; H0 (G)). Then for any solution of system (5.2), the following is true: 1 2 2 (uN , ZN ) ∈ L∞ F ([0, T ] × Ω; H0 (G)) × LF (Ω; L (0, T ; LQ )),

1 2 2 (ξ N , Z N ) ∈ L∞ F ([0, T ] × Ω; H0 (G)) × LF (Ω; L (0, T ; LQ )).


761

Moreover, sup kuN (t)k2H1 + sup kξ N (t)k2H 1 0

t∈[0,T ]

+E

Z

0

t∈[0,T ]

T 0

kZN (s)k2LQ ds + E

P-a.s. for some constant K, independent of N.

Z

0

T

kZ N (s)k2LQ ds ≤ K,

(5.5)

Proof. The proof is very similar to Proposition 3.1, with few modifications of some estimates. It is clear that s 4 3 27C 3 β 2hBN (uN (s)), uN (s)i ≤ kuN (s)k2L2 + kuN (s)k2H1 , 0 2β 4 and 2hg∇ξ N (s), uN (s)i ≤

4g2 N β kξ (s)k2L2 + kuN (s)k2H1 . 0 β 4

Thus under part one of Assumption (A.2), we have Z T Z β T N kuN (t)k2L2 + kZN (s)k2LQ ds + ku (s)k2H1 ds 0 2 t t s Z T 4 3 27C 3 4g2 N α+ kuN (s)k2L2 + ≤kφN k2L2 + kξ (s)k2L2 ds 2β β t Z T −2 hZN (s)dWN (s), uN (s)i. t

Since

2h∇ · (RN uN (s)), ξ N (s)i

4C12 β ≤C1 kuN (s)k2L2 + kuN (s)k2H1 + + C1 kξ N (s)k2L2 , 0 4 β

we have

Z β T N E Fr kuN (t)k2L2 + E Fr kξ N (t)k2L2 + ku (s)k2H1 ds 0 4 t Z T Z T + E Fr kZN (s)k2LQ ds + E Fr kZ N (s)k2LQ ds ≤E

Fr

t N 2 kφ kL2

+ E Fr

Z

t

T

t

+E

Fr

kψN k2L2

α+

s 3

4g2 4C12 27C34 + C1 kuN (s)k2L2 + + + C1 kξ N (s)k2L2 ds, 2β β β

for 0 ≤ r ≤ t, P-a.s. An application of the Gronwall inequality and letting r = t completes the proof. We skip the proof of part (ii) and (iii) since they are very similar to part (i) and the proof of Proposition 3.2. Note that the proof of (iii) uses the second half of the coercivity assumption.

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HONG YIN

Under our assumptions, the well-posedness of system (5.2) can be obtained similarly to Theorem 4.3. We shall skip the proof. Now we are ready to present the main result of this paper. Theorem 5.3. Suppose that the terminal conditions satisfy φ ∈ L∞ (Ω; H10 (G)) and ψ ∈ FT ∞ 1 LFT (Ω; H0 (G)). Then there exists an adapted solution (u, Z, ξ, Z) of system (5.1), such that 1 2 2 (u, Z) ∈ L∞ F ([0, T ] × Ω; H0 (G)) × LF (Ω; L (0, T ; LQ )),

1 2 2 (ξ, Z) ∈ L∞ F ([0, T ] × Ω; H0 (G)) × LF (Ω; L (0, T ; LQ )).

Proof. For technical reasons, let us introduce a new system. For any M1 ∈ N, M1 ≤ N, let R M1 be the projection of R to L2 M1 (G). Then clearly previous results on projected system (5.2) hold for  ∂uN M1 (t)   = −AuN M1 (t) − BN (uN M1 (t)) − g∇ξ N M1 (t) + f N (uN M1 (t)) + ZN M1 (t)dWN (t);     ∂ξ N∂tM1 (t) + ∇ · (R M1 uN M1 (t)) = Z N M1 (t)dW N (t);   ∂t    uN M1 (T ) = φN and ξ N M1 (T ) = ψN .

Let the unique adapted solution be (uN M1 , ZN M1 , ξ N M1 , Z N M1 ). First of all, let us establish several limits of convergent sequences. They are necessary when we perform the N M1 ∞ Galerkin approximation scheme. By Proposition 5.2, {uN M1 }∞ }N=1 , {ZN M1 }∞ N=1 , {ξ N=1 N M1 ∞ and {Z }N=1 are all uniformly bounded in respective spaces. Thus there exist u, ξ, Z, and Z, and a subsequence Nk , such that w

uNk M1 − →u

in L2F (Ω; L2 (0, T ; H10 (G))),

ξ Nk M1 − →ξ

2 in L∞ F ([0, T ] × Ω; L (G)),

w

w

ZNk M1 − →Z

in L2F (Ω; L2 (0, T ; LQ )),

Z Nk M1 − →Z

in L2F (Ω; L2 (0, T ; LQ )).

w

Since A is a continuous mapping from H10 (G) to H−1 (G), we know that kAukH−1 ≤ CkukH10 , for all u ∈ H10 (G) and some constant C. Thus combined with the assumptions on f, one gets w

AuNk M1 − f Nk (uNk M1 ) − → F1

in L2F (Ω; L2 (0, T ; H−1 (G))),

for some function F1 and some subsequence Nk . By Lemma 2.9,

1

3

kBN (uN M1 (t))kH−1 ≤ CG kBN (uN M1 (t))kL2 ≤ CG C3 kuN M1 (t)kL4 ≤ 2 4 CG2 C3 kuN M1 (t)kH10 . Thus w

BNk (uNk M1 ) − → F2

in L2F (Ω; L2 (0, T ; H−1 (G))),

for some function F2 and some subsequence Nk . For every t, let us define Lt : L2F (Ω; L2 (0, T ; LQ)) → L2F (Ω; L2 (0, T ; H−1 (G))) Z T J 7→ J(s)dW(s). t


763

Clearly Lt is a bounded linear operator. Hence it maps weakly convergent sequences to weakly convergent sequences, and Z T Z T w Nk M1 Nk Z (s)dW (s) − → Z(s)dW(s) in L2F (Ω; L2 (0, T ; H−1 (G))). t

t

Similarly, one can prove that Z Tn o w Z Tn o AuNk M1 (s) − f Nk (uNk M1 (s)) + BNk (uNk M1 (s)) ds − → F1 (s) + F2 (s) ds, in

t 2 LF (Ω; L2 (0, T ; H−1 (G)))

Z

t

and

T

t

w

Z Nk M1 (s)dW Nk (s) − →

One can also show that

T

Z

in L2F (Ω; L2 (0, T ; H −1(G))).

Z(s)dW(s)

t

2 2 2 −1 Lξ : L∞ F ([0, T ] × Ω; L (G)) → LF (Ω; L (0, T ; H (G))) Z T ξ 7→ ∇ξ(s)ds t

2 is a bounded linear operator. Since ξ Nk M1 ∈ L∞ F ([0, T ] × Ω; L (G)), we have Z T Z T w ∇ξ Nk M1 (s)ds − → ∇ξ(s)ds in L2F (Ω; L2 (0, T ; H −1(G))). t

t

Likewise, we have Z T Z w ∇ · (R M1 uNk M1 (s))ds − → t

T

in L2F (Ω; L2 (0, T ; H−1 (G))).

∇ · (R M1 u(s))ds

t

Thus we have shown that Z Tn Z o u(t) =φ + F1 (s) + F2 (s) + g∇ξ(s) ds − t

T

Z(t)dW(s),

(5.6)

t

and

ξ(t) = ψ +

Z

t

T M1

∇ · (R u(s))ds −

Z

T

Z(s)dW(s)

(5.7)

t

hold P-a.s. For notational convenience, let us denote Nk by N again. For any M2 ≤ N, let v ∈ L∞ ([0, T ] × Ω; H10 M2 (G)). Define F Z T 3 C14C54 13 4 r(t) , 2α + 5 K 3 ds, t 2 3 (ν − 1)κhC0 where n o n o∞ K = sup sup kukL4 ∪ sup kuN M1 kL4 + sup kvkL4 . (t,ω)∈[0,T ]×Ω

(t,ω)∈[0,T ]×Ω

N=1

(t,ω)∈[0,T ]×Ω

By Lemma 5.1, it is easy to see that

1 hAuN M1 (t) + BN (uN M1 (t)) − f N (uN M1 (t)) + r˙(t)uN M1 (t) 2 1 N N − Av(t) − B (v(t)) + f (v(t)) − r˙(t)v(t), R M1 uN M1 (t) − R M1 v(t)i ≤ 0. 2

764

HONG YIN

Integrating both sides and taking the expectation, one gets Z T 1 E e−r(s) hAuN M1 (s) + BN (uN M1 (s)) − f N (uN M1 (s)) + r˙(s)uN M1 (s), 2 0 R M1 uN M1 (s) − R M1 v(s)ids Z T 1 ≤E e−r(s) hAv(s) + BN (v(s)) − f N (v(s)) + r˙(s)v(s), R M1 uN M1 (s) − R M1 v(s)ids. 2 0 (5.8) √ An application of the Itô formula to e−r(s) k R M1 uN M1 (s)k2L2 yields Z T √ √ N 2 −r(0) N M1 2 M M 1 1 Ek R φ kL2 − Ee k R u (0)kL2 + 2E e−r(s) hg∇ξ N M1 (s), R M1 uN M1 (s)ids 0 Z T √ −E e−r(s) k R M1 ZN M1 (s)k2LQ ds 0 Z T √ =−E e−r(s) r˙(s)k R M1 uN M1 (s)k2L2 ds 0 Z T − 2E e−r(s) hAuN M1 (s) + BN (uN M1 (s)) − f N (uN M1 (s)), R M1 uN M1 (s)ids 0 Z T 1 = − 2E e−r(s) hAuN M1 (s) + BN (uN M1 (s)) − f N (uN M1 (s)) + r˙(s)uN M1 (s), 2 0 R M1 uN M1 (s)ids. (5.9) Applying the Itô formula to kξ N M1 (s)k2L2 to get Z T −E 2hg∇ · (R M1 uN M1 (s)), ξ N M1 (s)ids 0 Z T =E 2hg∇ξ N M1 (s), R M1 uN M1 (s)ids 0 Z T N 2 N M1 2 =gEkψ kL2 − gEkξ (0)kL2 − gE kZ N M1 (s)k2LQ ds. 0

Substituting (5.10) into (5.9), one gets √ √ Ek R M1 φN k2L2 − Ee−r(0) k R M1 uN M1 (0)k2L2 + gEkψN k2L2 − gEe−r(0) kξ N M1 (0)k2L2 Z T Z T √ −r(s) N M1 2 − gE e kZ (s)kLQ ds − E e−r(s) k R M1 ZN M1 (s)k2LQ ds 0 0 Z T 1 = − 2E e−r(s) hAuN M1 (s) + BN (uN M1 (s)) − f N (uN M1 (s)) + r˙(s)uN M1 (s), 2 0 M1 N M1 R u (s)ids.

By the lower semi-continuity of the norms, we have Z T 1 2 lim inf E e−r(s) hAuN M1 (s) + BN (uN M1 (s)) − f N (uN M1 (s)) + r˙(s)uN M1 (s), N→∞ 2 0 M1 N M1 R u (s)ids.

(5.10)


765

√ √ = − Ek R M1 φk2L2 + lim inf Ee−r(0) k R M1 uN M1 (0)k2L2 N→∞

− gEkψk2L2 + g lim inf Ee−r(0) kξ N M1 (0)k2L2 N→∞ Z T Z T √ −r(s) N M1 2 + g lim inf E e kZ (s)kLQ ds + lim inf E e−r(s) k R M1 ZN M1 (s)k2LQ ds N→∞ N→∞ 0 0 √ √ 2 −r(0) 2 2 M M 1 1 ≥ − Ek R φkL2 + Ee k R u(0)kL2 − gEkψkL2 + gEe−r(0) kξ(0)k2L2 Z T Z T √ + gE e−r(s) kZ(s)k2LQ + E e−r(s) k R M1 Z(s)k2LQ ds. (5.11) 0

0

√ Again applying the Itô formula to e−r(s) k R M1 u(s)k2L2 and kξ(s)k2L2 in (5.6) and (5.7) to get √ √ Ek R M1 φk2L2 − Ee−r(0) k R M1 u(0)k2L2 + gEkψk2L2 − gEe−r(0) kξ(0)k2L2 Z T Z T √ − gE e−r(s) kZ(s)k2LQ ds − E e−r(s) k R M1 Z(s)k2LQ ds 0 0 Z T 1 = − 2E e−r(s) hF1 (s) + F2 (s) + r˙(s)u(s), R M1 u(s)ids. (5.12) 2 0 Hence (5.11) and (5.12) imply Z T 1 2 lim inf E e−r(s) hAuN M1 (s) + BN (uN M1 (s)) − f N (uN M1 (s)) + r˙(s)uN M1 (s), N→∞ 2 0 M1 N M1 R u (s)ids. Z T 1 ≥2E e−r(s) hF1 (s) + F2 (s) + r˙(s)u(s), R M1 u(s)ids. (5.13) 2 0

Together with (5.8), one gets Z T 1 E e−r(s) hF1 (s) + F2 (s) + r˙(s)u(s), R M1 u(s) − R M1 v(s)ids 2 0 Z T 1 ≤ lim inf E e−r(s) hAv(s) + BN (v(s)) − f N (v(s)) + r˙(s)v(s), R M1 uN M1 (s) − R M1 v(s)ids N→∞ 2 0 Z T 1 = lim inf E e−r(s) hAv(s) + B(v(s)) − f(v(s)) + r˙(s)v(s), N→∞ 2 0 n o PN R M1 uN M1 (s) − R M1 v(s) ids. (5.14) w

Since uN M1 − → u in L2F (Ω; L2 (0, T ; H10 (G))), it is easy to show that n o w PN R M1 uN M1 − → R M1 u

in L2F (Ω; L2 (0, T ; H10 (G))) as well. Thus (5.14) becomes Z T 1 E e−r(s) hF1 (s) + F2 (s) + r˙(s)u(s), R M1 u(s) − R M1 v(s)ids 2 0 Z T 1 ≤E e−r(s) hAv(s) + B(v(s)) − f(v(s)) + r˙(s)v(s), R M1 u(s) − R M1 v(s)ids. 2 0

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HONG YIN

Since the above inequality holds for all v ∈ L∞ ([0, T ] × Ω; H10 M2 (G)) and all M2 ∈ N, we F know that it holds true for all v ∈ L∞ ([0, T ] × Ω; H10 (G)). Let us choose v=u + λw where F ∞ 1 w ∈ LF ([0, T ] × Ω; H0 (G)) and λ > 0, one gets Z T E e−r(s) hF1 (s) + F2 (s), R M1 wids 0 Z T 1 ≥E e−r(s) hAv(s) + B(v(s)) − f(v(s)) + λ r˙(s)w(s), R M1 w(s)ids. 2 0 Letting λ vanish to 0, and by the arbitrariness of w and the continuity of the coefficients, we know that F1 (s) + F2 (s) = Au(s) + B(u(s)) − f(u(s)) P-a.s. for all M1 ∈ N. The regularity of the solution is guaranteed by Proposition 5.2. The proof can then be completed by letting M1 go to infinity. 6. Uniqueness and Continuity In this section we deal with the uniqueness and continuity of the solution. Again we assume the uniform bound of the terminal conditions under H10 -norm. Such circumstances arise in certain other nonlinear stochastic partial differential equations such as stochastic Euler equations. Theorem 6.1. Suppose that the terminal conditions satisfy φ ∈ L∞ (Ω; H10 (G)) and ψ ∈ FT 1 L∞ FT (Ω; H0 (G)). Then system (5.1) admits a unique adapted solution (u, Z, ξ, Z) in 2 1 2 L∞ F ([0, T ] × Ω; H0 (G)) × LF (Ω; L (0, T ; LQ ))

1 2 2 ×L∞ F ([0, T ] × Ω; H0 (G)) × LF (Ω; L (0, T ; LQ )).

Moreover, the solution is continuous with respect to the terminal conditions in L∞ ([0, T ]; L2F (Ω; L2 (G))) × L2F (Ω; L2 (0, T ; LQ ))

×L∞ ([0, T ]; L2F (Ω; L2 (G))) × L2F (Ω; L2 (0, T ; LQ )).

Proof. Suppose that (u1 , Z1 , ξ1 , Z1 ) and (u2 , Z2 , ξ2 , Z2 ) are solutions of system (5.1) according to terminal conditions (φ1 , ψ1 ) and (φ2 , ψ2 ), respectively. Denote uˆ = u1 − u2 , Zˆ = Z1 − Z2 , Zˆ = Z1 − Z2 , ξˆ = ξ1 − ξ2 , φˆ = φ1 − φ2 , ψˆ = ψ1 − ψ2 . Then the differences satisfy  ∂u(t) ˆ  ˆ dW(t) ˆ  ˆ  ∂t = −Au(t) − B(u1 (t)) + B(u2 (t)) − g∇ξ(t) + f(u1 (t)) − f(u2 (t)) + Z(t) dt ;    ∂ξ(t) ˆ ˆ dW(t) ; ˆ = Z(t)   ∂t + ∇ · (Ru(t)) dt    u(T ˆ ˆ ) = φˆ and ξˆ(T ) = ψ.

Define

r(t) ,

Z

t

T

3 C14C54 13 4 2α + 5 K 3 ds, 2 3 (ν − 1)κhC0

(6.1)


767

where K=

sup (t,ω)∈[0,T ]×Ω

ku1 kL4 +

sup (t,ω)∈[0,T ]×Ω

ku2 kL4 .

√ 2 2 ˆ ˆ Then an application of the Itô formula to e−r(s) k Ru(s)k and kξ(s)k yields L2 L2 Z Z T T √ √ 2 2 ˆ ˆ 2 2 + gE ˆ ˆ 2L2 + gEkξ(t)k Ee−r(t) k Ru(t)k k Z(s)k ds + E e−r(s) k RZ(s)k L LQ ds L Q t t Z T √ ˆ 2 2 + gEkψk ˆ 2 2 + 2E ˆ + B(u1 (s)) − B(u2 (s)) − f(u1 (s)) + f(u2 (s)) e−r(s) hAu(s) =Ek Rφk L L t

√ ˆ 2 2 + gEkψk ˆ 22 . ≤Ek Rφk L L

1 ˆ ˆ + r˙(s)u(s), Ru(s)ids 2

Thus we have shown the uniqueness and continuity of solutions.

Now we have established the well-posedness of the backward stochastic tidal dynamics equation. Such well-posedness holds when the terminal conditions are uniformly bounded under H10 -norm. One may want to relax the conditions on terminal values to a weaker sense, such as uniformly boundedness in L2 sense. However, such problems are still open. The difficulty lies in the nonadaptiveness nature of the backward stochastic differential equations. For instance, the function r defined in Lemma 5.1 is not adaptive to the forward filtration. So in the proof of Theorem 5.3 and Theorem 6.1, we redefined r so that it is adaptive to the system. For this approach, we have to improve the regularity of the solution u appeared in the definition of r. Such obstacles do not arise in the forward stochastic systems. References 1. Adams, R. A. Sobolev Spaces, Academic Press, New York, Inc., 1975. 2. Agoshkov, V. I. Inverse problems of the mathematical theory of tides: boundary-function problem, Russ. J. Numer. Anal. Math. Modelling, 20, No. 1, 1–18 (2005). 3. Bismut, J. M. Conjugate Convex Functions in Optimal Stochastic Control, J. Math. Anal. Apl., 44, 384–404 (1973). 4. Breckner, H. Galerkin Approximation and the Strong Solution of the Navier-Stokes Equation, Journal of Applied Mathematics and Stochastic Analysis, 13:3(2000), 239–259. 5. Chow, P. L. Stochastic Partial Differential Equations, Taylor & Francis Group, Boca Raton, 2007. 6. Galilei, G. Dialogue Concerning the Two Chief World Systems, 1632. 7. Gallavotti, G. Foundations of Fluid Dynamics, Springer-Verlag, New York, Inc., 2002. 8. Hu, Y., Ma, J. and Yong, J. On semi-linear, degenerate backward stochastic partial differential equations, Probab. Theory Relat. Fields, 123, 381–411(2002). 9. Ipatova, V. M. Solvability of a Tide Dynamics Model in Adjacent Seas, Russ. J. Numer. Anal. Math. Modelling, 20, No. 1, 67–69 (2005). 10. Kallianpur, G. and Xiong, J. Stochastic Differential Equations in Infinite Dimensional Spaces, IMS Lecture Notes-Monograph Series, 26, 1995. 11. Kesavan, S. Nonlinear Functional Analysis. A First Course, Hindustan Book Agency, New Delhi, 2004. 12. Ladyzhenskaya, O. A. The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, 1969. 13. Lamb, H. Hydrodynamics, Dover Publications, New York, 1932. 14. Laplace, P. S., Recherches sur quelques points du système du monde, Mem. Acad. Roy. Sci, Paris, 88, 75–182 (1775).

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15. Lions, J. L. Sentinels and stealthy perturbations. Semicomplete set of sentinels, Mathematical and numerical aspects of wave propagation phenomena, 239–251, SIAM, Philadelphia, PA, 1991. 16. Lions, J. L. Distributed systems with incomplete data and problems of environment: Some remarks, Mathematics, climate and environment, 58–101, RMA Res. Notes Appl. Math., 27, Masson, Paris, 1993. 17. Ma, J., Protter, P. and Yong, J. Solving Forward-Backward Stochastic Differential Equations Explicitly–A Four Step Scheme, Prob. Th. & Rel. Fields, 98 (1994), 339–359. 18. Ma, J. and Yong, J. Adapted solution of a degenerate backward SPDE, with applications, Stoch. Proc. and Appl., 70, 59–84(1997). 19. Ma, J. and Yong, J. On linear, degenerate backward stochastic partial differential equations, Probab. Theory Relat. Fields, 113, 135–170(1999). 20. Maclaurin, C., De Causâ Physicâ Fluxus et Refluxus Maris, 1740. 21. Marchuk, G. I. and Kagan, B. A. Ocean Tides. Mathematical Models and Numerical Experiments, Pergamon Press, Elmsford, NY, 1984. 22. Marchuk, G. I. and Kagan, B. A. Dynamics of Ocean Tides, Kluwer Academic Publishers, Dordrecht/ Boston/ London, 1989. 23. Manna, U., Menaldi, J. L. and Sritharan, S. S. Stochastic Analysis of Tidal Dynamics Equation, Infinite Dimensional Stochastic Analysis, Special Volume in honor of Professor H-H. Kuo, World Scientific Publishers, 90–113, 2008. 24. Menaldi, J. L. and Sritharan, S. S. Stochastic 2-D Navier-Stokes Equation, Appl Math Optim, 46:31–53 (2002). 25. Newton, I. Philosophiae Naturalis Principia Mathematica, 1687. 26. Pardoux, E. Equations aux dérivées partielles stochastiques non linéaires monotones. Etude des solutions fortes de type Itô, Thèse, Université de Paris Sud. Orsay, Novembre 1975. 27. Pardoux, E. and Peng, S. Adapted Solution of a Backward Stochastic Differential Equation, Systems and Control Letters, 14, 55–61, 1990. 28. Poincaré, H. Lecons de Mécanique Celeste. 3. Théroe des marées, Cauthier-Villars, Paris, 469 p. 1910. 29. Prévôt, C. and Röckner, M. A concise Course on Stochastic Partial Differential Equations, Springer-Verlag, Berlin, 2007. 30. Sritharan, S. S. and Sundar P. Large Deviations for Two-Dimensional Navier-Stokes Equations with Multiplicative Noise, Stochastic Processes & Their Applications, 116 (2006), 1636–1659. 31. Sundar, P. and Yin, H. Existence and Uniqueness of Solutions to the Backward 2D Stochastic Navier-Stokes Equations, Stochastic Processes and Their Applications, 119 (2009) 1216–1234. 32. Temam, R. Navier-Stokes Equations, North-Holland Publishing Company, New York, Inc., 1979. 33. Thomson, W. and Tait, P. G. Treatise of Natural Philosophy, Vol. I, 1883. 34. Yong, J. and Zhou, X. Y. Stochastic Controls, Springer-Verlag, New York, Inc., 1999. 35. Zeidler, E. Nonlinear Functional Analysis and its Applications Vol.I, II/A, II/B, Springer-Verlag, New York, Inc., 1990. Hong Yin: Department of Mathematics, State University of New York, Brockport, Brockport, NY 14420 E-mail address: [email protected]

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