WEALTH OPTIMIZATION AND DUAL PROBLEMS

WEALTH OPTIMIZATION AND DUAL PROBLEMS FOR JUMP STOCK DYNAMICS WITH STOCHASTIC FACTOR

Claudia Ceci Dipartimento di Scienze, Facolta’ di Economia Universita’ di Chieti-Pescara, I-65127-Pescara, Italy

Anna Gerardi Dipartimento di Ingegneria Elettrica e dell’Informazione, Facolta’ di Ingegneria Universita’ dell’ Aquila, I-67100-L’ Aquila, Italy

Abstract An incomplete financial market is considered with a risky asset and a bond. The risky asset price is a pure jump process whose dynamics depend on a jump-diffusion stochastic factor describing the activity of other markets, macroeconomics factors or microstructure rules that drive the market. With a stochastic control approach, maximization of the expected utility of terminal wealth is discussed for utility functions of Constant Relative Risk Aversion (CRRA) type. Under suitable assumptions, closed form solutions for the value functions and for the optimal strategy are provided and Verification results are discussed. Moreover, the solution to the dual problems associated to the utility maximization problems are derived. Keywords: Utility Maximization, Pure Jump processes, Jump-diffusion processes. 2000 Mathematical Subject Classification: 91B70; 60J75; 91B16; 93E20.

1. Introduction We study a problem of portfolio selection in an incomplete financial market with a risky and a nonrisky asset. The price of the risky asset follows a geometric marked point process whose dynamics depend on a correlated stochastic factor. The stochastic factor may describe the activity of other markets, macroeconomics factors or microeconomics rules that influence the market. The motivation to study this class of optimal investment models comes from their wide applicability. The simplest model that fits in our framework is the one with nonlinear stock dynamics where the correlated stochastic factor can be identified with the nonlinear component or can represent a non-traded asset. In fact the effects of correlation are also important in pricing derivatives written on non-traded assets, which are often closely correlated to the available for trading underlying asset ([10] and references therein). We consider an agent with Constant Relative Risk Aversion (CRRA) preferences, including the logarithmic utility, that wants to maximize her/his expected utility of terminal wealth by investing in the bond and in the risky asset. We work in a Markovian setting and we treat the utility maximization problems by stochastic control methods. The contribution of this paper is to provide explicit solutions for the value functions and optimal investment strategies. The fundamental stochastic model of optimal investment was first introduced by Merton ([26]) who exhibited closed form solutions under the assumption that the stock price follows a geometric Brownian motion and for special utility functions, in particular of CRRA type. Optimal investment models have been extensively studied for diffusion price dynamics, by using stochastic control techniques or convex duality methods, see for example [23], [11], [24], [13], [32], [33], [28] and [4]. In particular, a general diffusion case, where the underlying stock price is a solution of a differential equation has been analyzed in [32] when the coefficients are non-linear functions of the current stock level, and in [33] and [28] when the coefficients depend on a correlated stochastic factor. In [22], [1], [7] and [5] the wealth optimization problem has been studied in markets driven by asset prices which may exhibit a jumping behaviour and in [6] in presence of a stochastic factor for exponential utility. 1

In [22] a complete market with jump-diffusion prices has been analyzed while the case of incomplete market is considered by [1]. In [5] a pure jump multidimensional incomplete market driven by independent Poisson processes is discussed. In the last quoted papers the assumption of constant coefficients of the underlying stock prices has been made. The novelty of this note consists in introducing a non-linear pure jump stock dynamics whose coefficients evolve according to a correlated jump diffusion factor. With the advent of intraday information on financial asset price quotes a recent research in finance has been devoted to the analysis and modelling of high-frequency data for asset prices. Since real asset prices, on a small time scale, are piecewise constant and jump only at discrete points in times in reaction to trades or to significant new information marked point models have been considered (see, for instance, [30], [29], [16], [17], and [9]). In some of the quoted papers, the local characteristics of the stock price process depend on a latent process which may be considered as a stochastic factor or as an unobservable state variable. All these models take into account the random fluctuations of market activity, which are related to fluctuations in the amount of incoming news. In this paper, we consider a pure jump unidimensional market driven by two doubly stochastic independent Poisson processes and whose jump sizes may depend on a stochastic factor X, which is described by a Markov jump-diffusion process. Moreover, the dynamics of the risky asset and the stochastic factor may be strongly dependent, in particular the two processes may have common jump times. This model could take into account the possibility of catastrophic events. This kind of events, in fact, influences both the asset prices and the hidden state variable which drives their dynamics. To the author’s knowledge it is the first time that the utility maximization problem is explicitly solved in such a model and for an agent with CRRA preferences. In [6] a similar jump model has been treated for an agent with exponential utility function. In [33] the case where the price of the stock and the stochastic factor are diffusion processes driven by two correlated Brownian motion has been studied. As in this paper we will consider the stochastic factor fully observed across time. We work in a Markovian setting and we treat the utility maximization problems by stochastic control methods ([26], [32], [33], [1], [28]). Other approaches are proposed in literature by using the convex duality theory ([2], [3], [31] [5] and references therein). The paper is organized as follows. The model is described in Section 2. In Section 3, we define the optimization problems and we write down the associated Hamilton-Jacobi-Bellman (HJB) equation. Making an ansatz for the value functions we reduce the associated HJB-equation to linear equations whose solutions can be represented by the Feymnan-Kac formulas and whose regularity is discussed in the Appendix. Thus we get explicit expressions for the value functions and optimal investment strategies. The logarithmic case is studied in Section 4 and the power law case in Section 5. In the power law case the closed form for the value function is written under a new probability measure P˜ equivalent to the real world probability measure P . Under P˜ the stochastic factor is a jump-diffusion process as under the probability P , but with a different intensity for the driving point process which takes into account common jump times with the risky asset. In both the cases, the optimal investment rules are Markovian, dependent on the current stochastic factor, and linear in the wealth variable. In particular, when the jump sizes of the underlying stock price and the intensities of the point processes which drive its dynamics are constant, the optimal strategy dictates to keep a fixed proportion of the current total wealth as in the Merton’s original problem with CRRA preferences. Section 5 is devoted to derive the solutions of the dual problems associated to the wealth optimization problems. The solutions to the utility maximization problems, obtained in Section 4 and 5 by stochastic control techniques, allow us to obtain explicit solutions to the associated dual problems. A discussion on verification results is performed in the Appendix, where, by applying the viscosity solution method, existence and uniqueness of smooth solutions to the linear second order parabolic integro-differential equations, arising in Section 4 and 5, are given. This latter result in turn implies existence and uniqueness of a smooth solution of the Hamilton-Jacobi-Bellman equation associated to the control problems introduced in Section 3. 2. Market Model We consider a finite time horizon investment model on [0, T ] with one riskless money market account and a risky asset. The price of the bond is taken equal to 1. The risky asset price, S is modeled as a geometric 2

marked point process, whose dynamics is driven by two independent doubly stochastic Poisson processes, describing upwards and downwards jumps. Moreover its dynamics depends on a stochastic factor X, modeled as a jump-diffusion process having common jump-times with S. Given, on an underlying probability space (Ω, F, P ), a Brownian motion Wt and three independent doubly stochastic Poisson processes, Nti , i = 0, 1, 2, independent of Wt . We assume that the process X satisfies, on [0, T ], the stochastic differential equation dXt = b(t, Xt ) dt + σ(t, Xt ) dWt + K0 (t, Xt− ) dNt + dNt0 )

X0 = x ∈ IR.

with

(2.1)

where Nt = Nt1 + Nt2 . Overall this paper we shall assume suitable conditions in order to obtain existence and uniqueness to the equation (2.1) (see Proposition 2.2 below). The stock price satisfies   dSt = St− K1 (t, Xt− )dNt1 − K2 (t, Xt− )dNt2

with

S0 = s0 ∈ IR+ .

(2.2)

Setting Ft = σ{Wu , Nui , i = 0, 1, 2, u ≤ t}, we suppose that the processes Nti , i = 0, 1, 2 and Nt admits {P, Ft }-intensities denoted by λi (t), i = 0, 1, 2 and λ(t) = λ1 (t) + λ2 (t) , respectively, which are bounded, positive, measurable functions. The IR-valued functions b(t, x), σ(t, x), Ki (t, x), λi (t) i = 0, 1, 2 are jointly measurable and Ki (t, x) > 0, i = 0, 1, 2. Let us observe that in this model the process N 0 counts the jumps of X when S does not jump. The processes N i , i = 1, 2 are introduced in order to describe the jumps times of S. Note that X and S are correlated since common jump times are allowed. More precisely the quadratic variation of X and S is Z t
[X, S]t = K0 (r, Xr− )Sr− K1 (r, Xr− )dNr1 − K2 (r, Xr− )dNr2 . (2.3) 0

From now on, we shall assume the inequalities A1 ≤ λi (t) ≤ A2 ,

A1 ≤ Ki (t, x) ≤ A2 ,

i = 1, 2

and

K2 (t, x) < 1

∀t ∈ [0, T ], x ∈ IR

(2.4)

with A1 , A2 positive constants. By the Dol´ eans-Dade exponential formula we get that St = S0 eYt .

(2.5)

and, the logreturn process Y can be written as Z t Z t
Yt = log (1 + K1 (r, Xr− ))dNr1 + log 1 − K2 (r, Xr− ) dNr2 . 0

(2.6)

0

In the next Propositions we will give the semimartingale structure for the risky asset St , and under some classical assumption, we will state the Markov property for the process X. Proofs uses standard arguments and will be omitted. Proposition 2.1 St is a special locally bounded semimartingale with the decomposition St = S0 + Mt + At where ASt

Z =

t

  Sr− K1 (r, Xr− )λ1 (r) − K2 (r, Xr− )λ2 (r) dr

0

is a predictable process with bounded variation paths, Z t Z t S 1 Sr− K1 (r, Xr− )(dNr − λ1 (r)dr) − Sr− K2 (r, Xr− )(dNr2 − λ2 (r)dr) Mt = 0

0

3

is a square-integrable martingale whose angle process is given by Z t   S Sr2− K1 (r, Xr− )2 λ1 (r) + K2 (r, Xr− )2 λ2 (r) dr. < M >t = 0

Proposition 2.2 Under the following assumptions, | b(t, x) |2 + | σ(t, x) |2 + | K0 (t, x) |2 ≤ C(1+ | x |2 ),

C>0 (2.7)

For any R > 0 there exists a constant LR > 0 such that when | x |< R, | y |< R | b(t, x) − b(t, y) | + | σ(t, x) − σ(t, y) | + | K0 (t, x) − K0 (t, y) |≤ LR | x − y | there exists a unique solution to equation (2.1) such that b + |x|2 ) IE[Xt2 ] ≤ C(1

(2.8)

b suitable positive constant. The solution is a Markov process whose generator is for C ∂f (t, x) + LX (2.9) t f (t, x) = ∂t   
∂f ∂f 1 ∂2f = (t, x) + b(t, x) (t, x) + σ 2 (t, x) 2 (t, x) + f t, x + K0 (t, x)) − f (t, x) λ0 (t) + λ(t) ∂t ∂x 2 ∂x LX f (t, x) =

3. Optimal investment problem. In a market where continuous trading and unlimited short selling are possible, an investor with initial capital t z0 > 0, invests at any time t ∈ [0, T ] the amount θt SS− in the risky asset, and his remaining wealth in the t

t is a self-financing trading strategy. bond. We assume that θt SS− t Thus, the dynamics of the wealth process controlled by the investment process θt , can be written as   θt dZt = dSt = θt K1 (t, Xt− )dNt1 − K2 (t, Xt− )dNt2 , Z0 = z0 . St−

(3.1)

For a given strategy θt , the solution process Zt to (3.1) will of course depend on the chosen investment policy θt . To be precise we should therefore denote the process Zt by Ztθ , but sometimes we will suppress θ. A strategy θt is said to be admissible if it is a IR-valued (P, Ft )-predictable process such that the following integrability condition is satisfied Z

T

|θt |dt < +∞ P − a.s.

(3.2)

0

and there exists a unique process satisfying (3.1) and verifying the condition IE|Zt | < +∞, t ∈ [0, T ], and the state constraint Zt > 0 a.e.t ∈ [0, T ]. We denote by Θ the set of admissible policies. e Xt− )Zt− satisfies Let us observe that the wealth associated to Markov control policies of the form θt = θ(t, dZt = Zt− dMt where Z Mt =

t

  e Xu− ) K1 (u, Xu− )dN 1 − K2 (u, Xu− )dN 2 . θ(u, u u

0

Hence, by the Dol´ eans-Dade exponential formula, Zt is well defined on t ∈ [0, T ] and by assumption (2.4) we get that the class of admissible investment strategies is not empty. 4

Lemma 3.1 The set of admissible investment strategies Θ contains the following set of Markovian policies n  e Xt− )Zt− : θ(t, e x) ∈ Θ1 = θt = θ(t,

o −1 1 . , K1 (t, x) K2 (t, x)

The wealth associated to such strategies is strictly positive and has the expression Z t 

 e Xs− )K2 (s, Xs− ) dN 2 . e Xs− )K1 (s, Xs− ) dN 1 + log 1 − θ(s, Zt = z0 exp log 1 + θ(s, s s

(3.3)

(3.4)

0

We consider an agent with a utility function of a Separable Constant Relative Risk Aversion (CRRA) type  α  Uα (t, x, z) = zα Gα (x) 0 < α < 1, (3.5)  Uα (t, x, z) = log z + G0 (x) α = 0. with Gα (x) bounded smooth function for 0 ≤ α < 1. Note that the classical CRRA utility function can be obtained with Gα = 1, 0 < α < 1 and G0 = 0. The choice made in this note has been suggested by [33]. The investor’s objective is to maximize the expected utility from terminal wealth h i IE Uα (T, XT , ZT ) .

(3.6)

By considering the utility maximization problem as a stochastic control problem with only final reward, we introduce the associated value function   Vα (t, x, z) = sup IE Uα (T, XT , ZT ) | Xt = x, Zt = z , θ∈Θ(t)

Θ(t) denoting the class of admissible strategies on [t, T ]. A classical approach in stochastic control theory consists in writing down the Hamilton-Jacobi-Bellman (HJB) equation that the value function is expected to satisfy. In general, we can say that the value function solves the HJB equation only when we know, a priori, that it has enough regularity. On the other hand, Verification results allow us to claim that, if there exists a function F (t, x, z) classical solution of the HJB equation and the supremum in it is attained at θ∗ (t, x, z), then F coincides with the value function and ∗ θt∗ = θ∗ (t, Xt− , Zt∗− ) is an optimal feedback control (here, and in the sequel, Z ∗ = Z θ ). The HJB equation, for the model presented in this note, is ∂u (t, x, z) + sup Lθt u(t, x, z) = 0 ∂t θ

t ∈ (0, T ), x ∈ IR, z > 0

(3.7)

with the final condition u(T, x, z) = Uα (T, x, z), where Lθ denotes the generator of the controlled Markov process (Xt , Zt ) associated to the constant strategy θ ∂f (t, x, z) + Lθt f (t, x, z) = (3.8) ∂t   2
∂f ∂f 1 ∂ f = (t, x, z) + b(t, x) (t, x, z) + σ 2 (t, x) 2 (t, x, z) + f t, x + K0 (t, x), z − f (t, x, z) λ0 (t) + ∂x 2 ∂x ∂t  

+ f t, x + K0 (t, x), z + θK1 (t, x) − f (t, x, z) λ1 (t) + f t, x + K0 (t, x), z − θK2 (t, x) − f (t, x, z) λ2 (t). Lθ f (t, x, z) =

In the next Sections we shall assume that the HJB equation admits a smooth solution. Under suitable hypotheses, in particular under (2.4), we shall get a closed form solutions for the value functions and for the optimal strategy. Verifications results will be discussed in Appendix. In particular we shall prove that the optimal strategy is a Markovian one of the form θt∗ = θe∗ (t, Xt− ) Zt∗− where θe∗ (t, x) is defined in the following Lemma. 5

 1 , Lemma 3.2 There exists a unique solution, θe∗ (t, x) ∈ ( K1−1 (t,x) K2 (t,x) , to the following equation ∀t ∈ [0, T ], x ∈ IR, 0 ≤ α < 1 e x)K1 (t, x))α−1 λ1 (t)K1 (t, x) − (1 − θ(t, e x)K2 (t, x))α−1 λ2 (t)K2 (t, x) = 0. (1 + θ(t,

(3.9)

Proof. It is sufficient to observe that, for any fixed t ∈ [0, T ], x ∈ IR, the function Φα (θ) = (1 + θK1 (t, x))α−1 λ1 (t)K1 (t, x) − (1 − θK2 (t, x))α−1 λ2 (t)K2 (t, x),   1 and is continuous, strictly decreasing in K1−1 , (t,x) K2 (t,x) lim

θ→ K −1 (t,x)

Φα (θ) = +∞,

lim1

θ→ K

1

0≤α 0, P − a.s. and U Observe that U Z

t

et = 1 + L

e s− L 0

2 X

esi (dNsi − λi (s)ds) U

i=1

e t is a (P, Ft )-martingale by we get that L IE

hZ 0

t

e s− L

2 X

i esi | λi (s)ds < +∞. |U

i=1

Finally(5.2) and (5.3) follow by Girsanov Theorem and Ito formula. Next get a closed form for the value function in the power law utility case. Proposition 5.2 When the class of admissible investment strategies reduces to Θ1 defined in (3.3), the associated value function is of the form Vα1 (t, x, z) =

zα h(t, x), α

where h(t, x) is a bounded function given by RT i h θ )(s,Xs )−λ(s))ds e Gα (XT ) e t (Hα (e h(t, x) = sup E | Xt = x , e θ

(5.5)

(5.6)

e x) is e denotes the expected value under Pe, the probability measure defined in Lemma 5.1 and Hα (θ)(t, E defined in (5.4). 8

Proof. Since the wealth associated to strategies belonging to Θ1 is given by (3.4) we get that Vα1 (t, x, z) = sup IE θ∈Θ1

h n zα sup IE Gα (XT )exp = α e θ

Z

 Zα T

α

 Gα (XT ) | Xt = x, Zt = Z =

(5.7)

T

o i 

e Xs− )K2 (s, Xs− ) dN 2 e Xs− )K1 (s, Xs− ) dN 1 +log 1−θ(s, | Xt = x , α log 1+θ(s, s s

t

then the value function is of the form (5.5) and the expression of h(t, x) can be deduced by (5.7). By assumptions (2.4) there exists a constant C > 0 such that Gα (XT )exp

nZ

T

o 

e Xs− )K2 (s, Xs− ) dN 2 e Xs− )K1 (s, Xs− ) dN 1 + log 1 − θ(s, ≤ eC(T +NT ) α log 1 + θ(s, s s

t

which implies that h(t, x) is a bounded function. Finally, notice that h

E Gα (XT ) exp

nZ

T

i
o 
e Xs− )K1 (s, Xs− ) dN 1 + log 1 − θ(s, e Xs− )K2 (s, Xs− ) dN 2 | Xt = x = α log 1 + θ(s, s s

t

=E

hL e

T

et L

RT RT  h i (H (e θ )(s,Xs )−λ(s))ds θ )(s,Xs )−λ(s))ds e Gα (XT ) e t (Hα (e Gα (XT ) e t α | Xt = x = E | Xt = x ,

e denotes the expected value under the probability measure Pe defined in Lemma 5.1. Hence (5.6) where E follows. Theorem 5.3 If there exists a classical solution h(t, x) to the linear equation with terminal condition
∂h (t, x) + Le∗t f (t, x) + Hα (θe∗ )(t, x) − λ(t) h(t, x) = 0, ∂t

h(T, x) = Gα (x),

(5.8)

where • θe∗ (t, x) is given in Lemma 3.2 with α ∈ (0, 1), •

e x) = θe∗ (t, x), the operator Le∗t is obtained by the operator defined in (5.3), setting θ(t,

then the value function is given by Vα (t, x, z) =

zα h(t, x), α

(5.9)

and θt∗ = θe∗ (t, Xt− )Zt∗− ∈ Θ1 , is an optimal investment strategy. Moreover the following Feynman-Kac representation holds RT h i θ ∗ )(s,Xs )−λ(s))ds e ∗ Gα (XT ) e t (Hα (e h(t, x) = E | Xt = x (5.10) e x) = e ∗ denotes the expected value under the probability measure Pe∗ defined in Lemma 5.1 for θ(t, where E ∗ e θ (t, x). Proof. Again we are looking for a classical solution to the HJB-equation, and we choose a candidate solution of the form zα Vα (t, x, z) = h(t, x) α Setting θe = zθ , θ control variable appearing in (4.8), we obtain by direct computations that h(t, x) solves
∂h e (t, x) + sup LeX t h(t, x) + Hα (θ)(t, x) − λ(t) h(t, x) = 0, ∂t e θ The equation in (5.11) can be written as

9

h(T, x) = Gα (x),

(5.11)


 ∂h ∂h 1 ∂2h (t, x) + b(t, x) (t, x) + σ 2 (t, x) 2 (t, x) + λ0 (t) h t, x + K0 (t, x)) − h(t, x) + ∂t ∂x 2 ∂x
e x) = 0. −λ(t)h(t, x) + h t, x + K0 (t, x) sup Hα (θ)(t, e θ We find that the maximum is achieved at θe∗ (t, x) defined in Lemma 3.2, since zα α h(t, x)

dHα (e θ) de θ

e with α ∈ (0, 1). = Φα (θ)

is a classical solution to the HJB-equation. Thus (5.11) reduces to (5.8) and Vα (t, x, z) = By Verification results Vα (t, y, z) is the value function and θt∗ = θe∗ (t, St− )Zt∗− ∈ Θ1 is an optimal investment strategy. Finally, by applying Feynman-Kac formula we have the representation (5.10). 6. Duality In this Section we provide the solution to the dual problems associated to the utility maximization problems discussed in this note. In the sequel we will set G0 = 0 and, for 0 < α < 1, Gα = 1. According to the theory of convex duality the following duality relation holds ([25])  h  h i i 0 sup IE Uα (ZT ) = 0inf inf γz0 + IE Ψα γLP , T P ∈Me γ>0

θ

(6.1)

where Ψα is the conjugate convex function associated to Uα , defined by Ψα (y) = sup [Uα (x) − yx] y > 0. x∈IR

For the utility functions

zα α

 

Uα (z) =



Uα (z) = log z

the conjugate functions are given by  

Ψα (y) =

0 < α < 1, ,

α 1−α α−1 , α y

0 0, L

(7.8)

If moreover the function G0 introduced in (3.5) is such that e − y| |G0 (x) − G0 (y)| ≤ L|x

then the function h(t, x) defined in (4.11) is a bounded continuous function. 13

Proof. To prove (7.7) it is sufficient to observe that H0 (θe∗ )(t, x) can be written as
H0 (θe∗ )(t, x) = λ(t) log K1 (t, x) + K2 (t, x) − λ1 (t) log K2 (t, x) − λ2 (t) log K1 (t, x) + ϕ(t) and that, for any η1 , η2 ∈ [A, B], Z | log η1 − log η2 | =

η2

η1

1 1 dξ ≤ |η1 − η2 | ξ A

thus the assertion, recalling (2.4). As far as the claimed properties of the function given in (7.3) we note that it is bounded by definition, and that, by (7.8), (7.7) and Lemma 7.1 e h |x − y|. |h(t, x) − h(t, y)| ≤ L

(7.9)

Furthermore, the Markov property of the process Xt allow us to write, for s < t h i hZ t i h(s, x) = IE h(t, Xts,x ) + IE H0 (θe∗ )(r, Xrs,x ) dr

(7.10)

s

which in turn implies, by the second of (7.1), e h C(t e − s)1/2 (1 + |x|) + IE |h(t, x) − h(s, x)| ≤ L

hZ

t

H0 (θe∗ )(r, Xrs,x ) dr

i

s

that, joint with (7.9) guarantees the continuity. Then, as it is usual in this kind of discussion, we introduce the notion of viscosity solutions (see [12], [27], [8] and references therein). Definition 7.3 A continuous function h(t, x) is a viscosity solution to (7.4) when, for any (t, x) ∈ [0, T ]×IR, for any function φ ∈ Cb1,2 ([0, T ] × IR) such that h(t, x) = φ(t, x) - if h(s, y) > φ(s, y), for every (s, y) = / (t, x)   ∂φ ∗ e − (t, x) + LX φ(t, x) + H ( θ )(t, x) ≥0 0 t ∂t - if h(s, y) < φ(s, y), for every (s, y) = / (t, x),   ∂φ X ∗ e − (t, x) + Lt φ(t, x) + H0 (θ )(t, x) ≤ 0 ∂t

(supersolution).

(subsolution).

Lemma 7.4 Under the hypotheses of Lemma 7.2, and assuming the joint continuity of b(t, x), σ(t, x) and λi (t), Ki (t, x) i = 0, 1, 2, the function h(t, x) defined in (7.3) is a viscosity solution to (7.4). Proof. As in (7.10), for v > t we get v

h i hZ h(t, x) = IE h(v, Xvt,x ) + IE t

i h i hZ t,x H0 (θe∗ )(r, Xrt,x ) dr ≥ I E φ(v, X ) + I E − v

v

H0 (θe∗ )(r, Xrt,x ) dr

t

and, by Ito Formula IE

h

i

φ(v, Xvt,x )

Z = φ(t, x) + t

v



 ∂φ t,x X t,x IE (r, Xr ) + Lt φ(r, Xr ) dr. ∂r 14

i

Hence

Z t

v



 ∂φ t,x X t,x ∗ t,x e IE (r, Xr ) + Lt φ(r, Xr ) + H0 (θ )(r, Xr ) dr < 0. ∂r

By a dominated convergence argument we can perform the limit for v → t+ and we obtain that h(t, x) is a viscosity supersolution, and a similar procedure imply that it is a subsolution. As a conclusion, we give the following Theorem, in which we perform analogous argument as in [27], Proposition 5.3. Theorem 7.5 Assume (2.7), (7.2) and the hypotheses of Lemmas 7.2 and 7.4. In addition assume (i) (ii)

λi (t), Ki (t, x) i = 0, 1, 2, bounded, b(t, x), σ(t, x) bounded, locally Lipschitz in (t, x) and σ 2 (t, x) ≥ a > 0.

Then, the function h(t, x) given in (7.3) is the unique C 1,2 ((0, T ) × IR) ∩ C 0 ([0, T ] × IR) solution to the problem (7.4). Proof. Let us set the problem   ∂f 1 2 ∂2f ∂f (t, x) + b(t, x) (t, x) + σ (t, x) 2 (t, x) = Fh (t, x) − ∂t ∂x 2 ∂x with

f (T, x) = G0 (x).

(7.11)


 Fh (t, x) = λ0 (t) + λ(t) h(t, x + K0 (t, x)) − h(t, x) + H0 (θe∗ )(t, x).

The differential operator in the l.h.s. of (7.11) is a uniformly parabolic operator, by all the assumptions made on b(t, x) and σ(t, x), and Lemma 7.2 assures that Fh is a continuous function, Lipschitz in x uniformly in t. Thus, according to Theorem 5.3 in [18], we get the existence of a solution fh ∈ C 1,2 ((0, T )×IR)∩C 0 ([0, T ]×IR) to (7.11) verifying a sublinear growth condition. Since fh is also a bounded viscosity solution to the same problem, then, by a uniqueness result of viscosity solutions with sublinear growth, given in [20] it coincides with h which turns to be a classical solution to (7.4). 7.2. The power law case The discussion we will perform in this Subsection follows the same lines as that one given in the logarithmic case. According to Theorem 6.2 we have to prove existence of classical solutions to the problem (5.8). The main difference relies on the fact that, in this case we have to work under the measure Pe∗ , defined in e x) = θe∗ (t, x) (θe∗ given in Lemma 3.2). This implies that the process Nt has {Pe∗ , Ft }Lemma 5.1 for θ(t, intensity given by Hα (θe∗ )(t, Xt− ), Hα (θe∗ )(t, x) defined in (5.4) as
α
α Hα (θe∗ )(t, x) = 1 + θe∗ (t, x)K1 (t, x) λ1 (t) + 1 − θe∗ (t, x)K2 (t, x) λ2 (t)

(7.12)

Thus the process Xt is still a Markov process under Pe∗ , with generator Le∗t given by (5.3) and the candidate solution to the problem (5.8) admits the representation (5.10). Since Hα (θe∗ )(t, x) is bounded, we can write analogous inequalities as in (7.1), and in particular under (2.7) and (7.2) b (t − r)1/2 (1 + |x|) IeE ∗ [|Xtr,x − x|] ≤ C (7.13) IeE ∗ |Xtr,x − Xtr,y | ≤ Cα |x − y| Let us recall that the problem (5.8) can be written as
∂h (t, x) + Le∗t f (t, x) + Hα (θe∗ )(t, x) − λ(t) h(t, x) = 0, ∂t

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h(T, x) = Gα (x),

(7.14)

whose candidate solution admits the representation RT h i θ ∗ )(s,Xst,x )−λ(s))ds e ∗ Gα (X t,x ) e t (Hα (e , h(t, x) = E T

(7.15)

and let us notice the strict formal analogy with the problem discussed in the Appendix of [10]. At this point we need the following technical result Lemma 7.6 Assuming Ki (t, x), i = 1, 2 globally Lipschitz w.r.t. x, uniformly in t |Ki (t, x) − Ki (t, y)| ≤ LK |x − y|

x, y ∈ IR,

i = 1, 2

LK > 0,

(7.16)

the same global Lipschitz property holds for Γ(t, x), θe∗ (t, x) and Hα (θe∗ )(t, x). In particular |Hα (θe∗ )(t, x) − Hα (θe∗ )(t, y)| ≤ LH |x − y|

x, y ∈ IR,

LH > 0.

Proof. Being Ki (t, x), i = 1, 2 globally Lipschitz and bounded, the Lipschitz property of Γ(t, x) can be easily deduced. Then we write
1 θe∗ (t, x) = G Γ(t, x) K1 (t, x) with 1−



y+



G(t, y) =

λ2 (t) λ1 (t)

y

λ2 (t) λ1 (t)

y

1/(1−α) 1/(1−α)

and

∂ G(t, y) < const. ∂y

Finally

Hα (θe∗ )(t, x) = λ1 (t) F1 θe∗ (t, x)K1 (t, x) + λ2 (t) F2 θe∗ (t, x)K2 (t, x) with α

F1 (t, y) = (1 + y)

α

F2 (t, y) = (1 − y)

and

∂ Fi (t, y) < const, ∂y

i = 1, 2.

Furthermore, the Pe∗ -markovianity of Xt , allows us to write, for t < v Rv h i (H (e θ ∗ )(s,Xst,x )−λ(s))ds h(t, x) = IeE h(v, Xvt,x ) e t α . The last results, joint with (7.13), are the tools we need to get the following Lemma. Lemma 7.7 Under (2.7), (7.2), (7.16) and |Gα (x) − Gα (y)| ≤ Lα |x − y|

Lα > 0,

if b(t, x), σ(t, x), K0 (t, x) and λi (t), i = 0, 1, 2 are continuous functions the function h(t, x) is a bounded continuous function and it is a viscosity solution to (7.14). The proof is quite similar to these of Lemma 7.2 and Lemma 7.4 as well as the proof of the following concluding Theorem which is the analogous of Theorem 7.5. Theorem 7.8 Assume the hypotheses of Lemma 7.7. In addition assume (i) (ii)

b(t, x), σ(t, x), Ki (t, x) and λi (t), i = 0, 1, 2 bounded, b(t, x), σ(t, x) locally Lipschitz in (t, x) and σ 2 (t, x) ≥ a > 0.

Then, the function h(t, x) given in (7.15) is the unique C 1,2 ((0, T ) × IR) ∩ C 0 ([0, T ] × IR) solution to the problem (7.14).

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