Hyperbolic stochastic partial differential equations

Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet Mohamed Erraoui∗ Facult´e des Sciences Semlalia D´epartement de Math´ematiques Universit´e Cadi Ayyad BP 2390, Marrakech, Maroc

David Nualart† Facultat de Matem`atiques Universitat de Barcelona Gran Via, 585 08007, Barcelona, Spain

Youssef Ouknine‡, Facult´e des Sciences Semlalia D´epartement de Math´ematiques Universit´e Cadi Ayyad BP 2390, Marrakech, Maroc

Abstract 0 {BzH,H , z 0

2

Let ∈ [0, T ] } be a fractional Brownian sheet with Hurst parameters H, H ≤ 12 . We prove the existence and uniqueness of a strong solution for a class of hyperbolic stochastic partial differential equation 0 with additive fractional Brownian sheet of the form Xz = x + BzH,H + R b(ζ, Xζ )dζ, where b(ζ, x) is a Borel function satisfying some growth [0,z] and monotonicity assumptions. We also prove the convergence of Euler’s approximation scheme for this equation. Keywords: Fractional Brownian sheet. Hyperbolic stochastic partial differential equations. Stochastic integrals. Euler’s scheme. AMS Subject Classification: 60H05, 60H15.

1

Introduction

The motivation for the present work is to study the following hyperbolic stochastic partial differential equation  H,H 0  ∂ 2 u (s, t) ∂ 2 B(s,t) = b(s, t, u (s, t)) + , s, t ∈ [0, T ] (1) ∂s∂t ∂s∂t  u (s, 0) = u (0, t) = a, ∗ Supported

by Moroccan Program PARS MI 37 by the DGES grant BFM2000-0598 ‡ Supported by Moroccan Program PARS MI 37 † Supported

1

0

0

where B H,H = {BzH,H , z ∈ [0, T ]2 } is a fractional Brownian sheet with Hurst 1 0 parameter (H, H 0 ) ∈ (0, )2 . That is, B H,H is a centered Gaussian process 2 with covariance 0

0

)o n RH,Hn0 (z, z 0 ) = E(BzH,H BzH,H 0 o 0 2H 2H 0 2H 0 1 2H 0 2H 0 = 4 s + (s ) − |s − s| t2H + (t0 ) − |t0 − t| , where z = (s, t) and z 0 = (s0 , t0 ). The nonlinear coefficient is a Borel function b : [0, T ]2 × R → R. The main results of the paper are: (i) If b satisfies the linear growth condition: |b(z, x)| ≤ C(1 + |x|),

(2)

then Equation (1) has a unique weak solution. (ii) If the function b is nondecreasing with respect to the second variable and bounded, then Equation (1) has a unique strong solution. These results represent an extension to the two-parameter case of those proved in a recent work by Nualart and Ouknine [14] for the equation Z H b(s, Xs )ds, Xt = x + Bt + [0,t]

where BtH is 1-parameter fractional Brownian motion. The result (ii) in the case 0 H = H 0 = 12 , i.e.when B H,H is an ordinary Brownian sheet, has been proved by Nualart and Tindel in [15]. We also show the convergence of the Euler’s approximation scheme for Equation (1) under the assumptions given in (ii). A similar approximation result has been established for several types of stochastic differential equations (see [4], [9], [10] and references therein). A brief outline of the paper is as follows. In Section 2 we give some preliminaries on two-parameter fractional calculus. It is well-known that the main approach for constructing weak solutions to stochastic differential equations is the transformation of drift via Girsanov theorem. Since the fractional Brownian sheet has an integral representation of the form Z 0 BzH,H = KH,H 0 (z, ζ)dWζ , [0,z]

where W is a standard Brownian sheet and KH,H 0 is a square integrable kernel, the weak existence and uniqueness are easily established using a suitable version of Girsanov theorem. This is done in Section 3. In Section 4, a comparison theorem based on the monotonicity of the coefficient is given. In Section 5, the strong existence and uniqueness results are established. The proof is based on the comparison theorem established in Section 5 and the equi-absolute continuity of the occupation measures of a certain class of approximations of u. Note 2

that this method has also been used to handle one-dimensional heat equations with additive space-time white noise in [8]. Finally, in Section 6, using the result of Section 5, we prove the convergence of Euler’s approximation scheme for Equation (1).

2 2.1

Preliminaries Fractional calculus

In preparation for the following sections, we need to recall some basic definitions and results about two-parameter fractional calculus. An exhaustive survey on classical one-parameter ¡ ¢ fractional calculus can be found in [18]. For f ∈ L1 [0, T ]2 and α > 0 , β > 0 the left-sided fractional RiemannLiouville integral of f of order α, β on (0, T )2 is given at almost all (x, y) by Z xZ y 1 α,β I f (x, y) = (x − u)α−1 (y − v)β−1 f (u, v)dudv, (3) Γ(α)Γ(β) 0 0 where Γ denotes the Euler function. We remark that (3) can be written as follows: ¡ ¢ I α,β f (x, y) = I α I β f (·, y) (x) . This integral extends the usual (n, m)-order iterated integrals of f for α = (n, m) ∈ N2 . We have the first composition formula 0

0

0

0

I α,β (I α ,β f ) = I α+α ,β+β f. The fractional derivative can be introduced as the inverse operation. We assume 0 < α, β < 1 and p > 1. We denote by I α,β (Lp ) the image of Lp ([0, T ]2 ) by the operator I α,β . If f ∈ I α,β (Lp ), the function φ such that f = I α,β φ is unique in Lp and it agrees with the left-sided Riemann-Liouville derivative of f of order (α, β) defined by Z xZ y 1 ∂2 f (u, v) Dα,β f (x, y) = dudv. Γ(1 − α)Γ(1 − β) ∂x∂y 0 0 (x − u)α (y − v)β ´ ³ ´´ ³³ When inf (αp, βp) > 1 any function in I α,β (Lp ) is min α − p1 , β − p1 H¨older continuous. On the other hand, any H¨older continuous function of order (γ, δ) with γ > α and δ > β has a fractional derivative of order (α, β). The derivative of f has the following Weil representation: · Z x 1(0,T )2 (x, y) f (x, y) α f (x, y) − f (u, y) α,β D f (x, y) = + β du Γ(1 − α)Γ(1 − β) xα y β y (x − u)α+1 0 β + α x

Z

y 0

Z

f (x, y) − f (x, v) dv (y − v)β+1 x

Z

y

+ αβ 0

0

¸ f (x, y) − f (x, v) − f (u, y) + f (u, v) dudv , (x − u)α+1 (y − v)β+1 3

where the convergence of the integrals at the singularity x = u or y = v holds in Lp -sense. Recall that by construction for f ∈ I α,β (Lp ), I α,β (Dα,β f ) = f and for general f ∈ L1 ([0, T ]2 ) we have Dα,β (I α,β f ) = f. 0

0

If f ∈ I α+α ,β+β (L1 ), α ≥ 0, β ≥ 0, α0 ≥ 0, β 0 ≥ 0 , sup(α + β, α0 + β 0 ) ≤ 1 we have the second composition formula 0

0

0

0

Dα,β (Dα ,β f ) = Dα+α ,β+β f.

3

Existence of a weak solution, and pathwise uniqueness property

In this section we introduce the notion of weak solution for a stochastic differential equation with additive fractional Brownian sheet. We discuss the existence and uniqueness of such solutions and the uniqueness in law for such equation. For the proof of the main results we follow the approach developed by Nualart and Ouknine in [14] for the one-parameter case, based on Girsanov theorem.

3.1

Girsanov transformation 2

Let (Ω, F, P ) be a probability space. If z1 , z2 ∈ [0, T ] , zi = (si , ti ), i = 1, 2, we will write z1 ≥ z2 if s1 ≥ s2 and t1 ≥ t2 . For any z1 ≥ z2 , we set [z1 , z2 ] = 0 [s1 , s2 ] × [t1 , t2 ]. Let B H = {BzH,H , z ∈ [0, T ]2 } be a fractional Brownian 1 sheet with Hurst parameters 0 < H, H 0 ≤ defined on the probability space 2 2 (Ω, F, P ). For each we denote by FzB the σ-field generated by the n z ∈ 0[0, T ] o H,H random variables Bζ , ζ ≤ z and the sets of probability zero. We will say © ª that a stochastic process u = uz , z ∈ [0, T ]2 is FzB -adapted if u(z) is FzB measurable for all z ∈ [0, T ]2 . Given z1 = (s1 , t1 ) ≤ z2 = (s2 , t2 ) and a process X we denote by 4[z1 ,z2 ] (X) = X (s2 , t2 ) − X (s2 , t1 ) − X (s1 , t2 ) + X (s1 , t1 ) the increment of X on the rectangle [z1 , z2 ]. We denote by E ⊂ H the set of step functions on [0, T ]2 . Let H be the Hilbert space defined as the closure of E with respect to the scalar product ­ ® 1[0,z] , 1[0,z0 ] H = RH,H 0 (z, z 0 ). 0

The mapping 1[0,z] −→ BzH,H can be extended to an isometry between H 0 0 and the Gaussian space H1 (B H,H ) associated with B H,H . We will denote this 0 isometry by ϕ −→ B H,H (ϕ). 4

The covariance kernel RH,H 0 (z, z 0 ) can be written as Z 0 RH,H 0 (z, z ) = KH,H 0 (z, ζ)KH,H 0 (z 0 , ζ)dζ, [0,z∧z 0 ]

where [0, z ∧ z 0 ] = [0, s ∧ s0 ]×[0, t ∧ t0 ] and KH,H 0 is the square integrable kernel given by : KH,H 0 (z, z 0 ) = KH (s, s0 )KH 0 (t, t0 ), where KH , KH 0 are defined by (see [3]):  ¢ ¡ s0  H− 12 1 1 1 1 −1 0 0  (s − s) F (H − , − H, H + , 1 − ) K (s, s ) = Γ H +  H 2 2 2 2  s  0  ¡ ¢ 1   KH 0 (t, t0 ) = Γ H + 1 −1 (t0 − t)H− 2 F (H − 1 , 1 − H, H + 1 , 1 − t ), 2 2 2 2 t F (a, b, c, z) being the Gauss hypergeometric function. Consider the linear oper2 2 ∗ ator KH,H 0 from E to L ([0, T ] ) defined by Z ∗ (KH,H 0 ϕ)(z)

=K

H,H 0

((T, T ) , z)ϕ(z)+

(ϕ(ζ) − ϕ(z)) [z,(T,T )]

∂ 2 KH,H 0 (ζ, z)dζ. ∂ζ

For any pair of step functions ϕ and ψ in E we have ® ­ ∗ ∗ = hϕ, ψiH . KH,H 0 ϕ, KH,H 0ψ L2 ([0,T ]2 ) This is an immediate consequence of the fact that ¢ ¡ ∗ KH,H 0 1[0,z] (ζ) = KH,H 0 (z, ζ). ∗ As a consequence, the operator KH,H 0 provides an isometry between the Hilbert 2 2 spaces H and L ([0, T ] ). Hence, the process W = {Wz , z ∈ [0, T ]2 } defined by

¢−1 0 ¡ ∗ (1[0,z] )) Wz = B H,H ( KH,H 0

(4)

0

is a Brownian sheet, and the process B H,H has an integral representation of the form Z 0 BzH,H = KH,H 0 (z, ζ)dWζ . (5) [0,z]

We refer to [1] for the proof of similar results in the one-parameter case. From (4) and (5) it follows that FzB coincides with the filtration FzW generated by the Brownian sheet W . This filtration satisfies the following properties (see Cairoli and Walsh [2]): (F1) FzW is increasing with respect to the partial order in [0, T ]2 . W (F2) FzW is right-continuous, that is, for all z ∈ [0, T )2 we have ∩n Fz+( = 1 1 , ) n n

FzW . 5

(F3) F0W contains the sets of probability zero. W W (F4) For any 0 ≤ s, t ≤ T , the σ-fields ∨0≤u≤T F(u,t) and ∨0≤v≤T F(s,v) are W conditionally independent given F(s,t) .

We will make use of the following definitions. © ª Definition 1 Let Fz , z ∈ [0, T ]2 be a family of σ-fields satisfying properties (F1)-(F4). A Brownian sheet W = {Wz , z ∈ [0, T ]2 } is called an Fz -Brownian sheet if W is Fz -adapted and for any z1 ≤ z2 the increment 4[z1 ,z2 ] (W ) is independent of the σ-field generated by {W(u,v) , u ≤ s1 or v ≤ t1 }. © ª Definition 2 Let Fz , z ∈ [0, T ]2 be a family of σ-fields satisfying proper0 0 ties (F1)-(F4). A fractional Brownian sheet B H,H = {B H,H , z ∈ [0, T ]2 } is called an Fz -fractional Brownian sheet if the process W defined in (4) is an Fz -Brownian sheet. The operator KH,H 0 , on L2 ([0, T ]2 ) associated with the kernel KH,H 0 is an 0 isomorphism from L2 ([0, T ]2 ) onto I H+1/2,H +1/2 (L2 ([0, T ]2 )) and it can be expressed in terms of fractional integrals as follows (see [3] for the one-parameter case): 0

1

0

1

1

0

1

1

(KH,H 0 h)(s, t) = I 2H,2H s 2 −H t 2 −H I 2 −H, 2 −H sH− 2 tH

0

− 12

h,

(6)

where h ∈ L2 ([0, T ]2 ). Given a process with integrable trajectories u = {uz , z ∈ [0, T ]2 } consider the transformation Z ezH,H 0 = BzH,H 0 + B uζ dζ. (7) [0,z]

We can write ezH,H 0 = B

Z

Z

Z

KH,H 0 (z, ζ)dWζ + [0,z]

where

[0,z]

Z fz = Wz + W [0,z]

fζ , KH,H 0 (z, ζ)dW

uζ dζ =

Ã

[0,z]

ÃZ

−1 KH,H 0

! uζ dζ

! (η) dη.

(8)

[0,·]

´ ³R −1 Notice that KH,H u dζ belongs to L2 ([0, T ]2 ) almost surely if and only 0 [0,·] ζ R 0 if [0,·] uζ dζ ∈ I H+1/2,H +1/2 (L2 ([0, T ]2 )). As a consequence we deduce the following version of the Girsanov theorem for the fractional Brownian sheet: © ª Theorem 3 Let Fz , z ∈ [0, T ]2 be a family of σ-fields satisfying properties 0 0 (F1)-(F4). Suppose that B H,H = {B H,H , z ∈ [0, T ]2 } is an Fz -fractional Brownian sheet and W = {Wz , z ∈ [0, T ]2 } is the Fz -Brownian sheet given by (4). Consider the shifted process (7) defined by an Fz -adapted process u = {uz , z ∈ [0, T ]2 } with integrable trajectories. Assume that: 6

i)

R [0,·]

uζ dζ ∈ I H+1/2,H

0

+1/2

(L2 ([0, T ]2 )), almost surely.

ii) E(ξ T,T ) = 1, where ξ T,T

=

³ R ³ ³R ´´ −1 exp − [0,T ]2 KH,H u dζ (z) dWz 0 [0,·] ζ − 12

¶ ³ ³R ´´2 −1 KH,H 0 [0,·] uζ dζ (z) dz [0,T ]2

R

e H,H 0 is an Fz -fractional Brownian sheet with Hurst Then the shifted process B e dP parameter (H, H 0 ) under the new probability Pe defined by dP = ξ T,T . Proof. The filtration Fz satisfies the properties (F1)-(F4) under the new probability Pe (see [6]). By the extension of Girsanov theorem to twoparameter processes (see ´[6]) applied to the adapted and square integrable ³R −1 f defined in (8) is a process KH,H 0 [0,·] uζ dζ we obtain that the process W Fz -Brownian sheet under the probability Pe. Hence, the result follows. −1 From (6) the inverse operator KH,H 0 is given by 1

0

1

1

0

1

−1 2 −H t 2 −H D 2 −H, 2 −H s KH,H 0 h (s, t) = s 0

1

H− 21

t

H 0 − 12

0

D2H,2H h,

(9)

1

for all h ∈ I H+ 2 ,H + 2 (L2 ([0, T ]2 )). If h vanishes on the axes and is absolutely continuous, it can be proved that ∂2h . (10) ∂s∂t RT RT From (10) it follows that a sufficient condition for i) is 0 0 u2ζ dζ < ∞. 1

−1 H− 2 H t KH,H 0 h (s, t) = s

3.2

0

− 12

1

1

0

1

1

I 2 −H, 2 −H s 2 −H t 2 −H

0

Existence of a weak solution

We are going to use the Girsanov transformation (Theorem 3) in order to establish the existence of a weak solution for the following stochastic differential equation: Z 0

Xz = x + BzH,H +

b(ζ, Xζ )dζ, z ∈ [0, T ]2 ,

(11)

[0,z]

where b is a Borel function on [0, T ]2 × R. By ³a weak solution to equation (11) we mean a couple of Fz -adapted pro´ © ª H,H 0 cesses B , X on a filtered probability space (Ω, F, P, Fz , z ∈ [0, T ]2 ), such that the filtration Fz satisfies the properties (F1)-(F4) and: 0

i) B H,H is an Fz -fractional Brownian sheet in the sense of Definition 2. 0

ii) X and B H,H satisfy (11). 7

Theorem 4 Suppose that b(z, x) satisfies the linear growth condition (2). Then Equation (11) has a weak solution. e H,H 0 = B H,H 0 − Proof. Set B z z 0

R

0

[0,z]

b(ζ, BζH,H + x)dζ. We claim that the

process uζ = −b(ζ, BζH,H + x) satisfies conditions i) and ii) of Theorem 3. If ezH,H 0 is a FzB -fractional this claim is true, under measure Pe, B ³ the 0probability ´ 0 e H,H , B H,H + x is a weak solution of (11) on the Brownian sheet, and B z z ª © filtered probability space (Ω, F, Pe, FzB , z ∈ [0, T ]2 ). Set ÃZ ! 0

−1 vz = −KH,H 0

[0,.]

b(ζ, BζH,H + x)dζ

(z).

From (10) and the linear growth property of b we obtain ¯ ¯ 0 0 1 0 0 1 1 1 1 1 ¯ ¯ |vz | = ¯sH− 2 tH − 2 I 2 −H, 2 −H s 2 −H t 2 −H b(z, BzH,H + x)¯ =

≤

1

0

1

1 sH− 2 tH − 2 × 1 1 ¯RΓ( 2 −H)Γ( 2 −H 01) ¯ ¯ ¯ − 1 −H 0 1 −H 0 − −H 1 −H H,H 0 u2 (t − v) 2 v2 b((u, v) , B(u,v) + x)dudv ¯ ¯ [0,z] (s − u) 2 ° ° ³ ´ 0° ° cH,H 0 C 1 + |x| + °B H,H ° , ∞

(12) where cH,H 0 is a constant depending only on H , H 0 and T . From (10) it follows −1 that the operator (KH,H 0 ) preserves the adaptability property. Hence, the process vz is adapted and then condition ii) can be proved using a version of Novikov criterion for two-parameter processes. Indeed, it suffices to show that for some λ > 0 ³ ³ ´´ 2 sup E exp λ |vz | < ∞, (13) z∈[0,T ]2

which is an immediate consequence of (12) and the exponential integrability of the seminorm of a Gaussian process (see Fernique [5]).

3.3

³

Uniqueness in law 0

Let B H,H , X

´

be a weak solution of the stochastic differential equation (11) © ª defined in the filtered probability space (Ω, F, P, Fz , z ∈ [0, T ]2 ). Define ÃZ ! −1 uz = KH,H 0

b(ζ, Xζ )dζ

(z).

[0,.]

Let Pe defined by à Z ! Z dPe 1 2 = exp − uζ dWζ − u dζ dP 2 [0,T ]2 ζ [0,T ]2

8

(14)

We claim that the process uz satisfies conditions i) and ii) of Theorem 3. In fact, uz is an Fz -adapted process and taking into account that Xz has the same regularity properties than the fractional Brownian sheet we deduce that R 2 2 uζ dζ < ∞ almost surely. Finally, we can apply again Novikov theorem in [0,T ] ³ ´ e P order to show that E ddP = 1, because by Gronwall’s lemma for integrals in the plane ° ° ³ ´ 0° 2 ° kXk∞ ≤ |x| + °B H,H ° + CT 2 eCT . ∞

By Theorem 3 the process Z fz = Wz + W

uζ dζ [0,z]

fz is an Fz -Brownian sheet under the probability Pe. In terms of the process W we can write Z fz . Xz = x + KH,H 0 (z, ζ)dW [0,z]

Hence, then X − x is a fractional Brownian sheet with respect to the probability Pe with Hurst parameter equal to (H, H 0 ). As a consequence, we can easily prove e H,H 0 have the same distribution under as in [14] that the processes X − x and B the probability P . In conclusion we have proved the following result: Theorem 5 Suppose that b(z, x) satisfies the assumptions of Theorem 4. Then all weak solutions have the same distribution.

4

Comparison theorem

The standard comparison theorem does not hold for hyperbolic partial differential equations, as it is shown, for instance, in [13]. Nevertheless we can establish a comparison theorem for hyperbolic equations assuming the monotonicity of one of the coefficients: Theorem 6 Let u1 and u2 be two solutions of the deterministic equations Z ui (z) = x + β z + bi (ζ, ui (ζ))dζ, z ∈ [0, T ]2 , (15) [0,z]

i = 1, 2, where β z is a continuous function vanishing on the axes. Assume that 2

(j) For each z ∈ [0, T ] , b2 (z, x) is a nondecreasing function of x (jj) |b2 (z, x) − b2 (z, y)| ≤ ρ (|x − y|) , for all z ∈ [0, T ]2 (jjj) b1 (z, x) ≤ b2 (z, x), where ρ is concave, continuous, with ρ (0) = 0 and Z du = +∞. ρ (u) + 0 9

Then the solutions u1 and u2 satisfy u1 (z, x) ≤ u2 (z, x). Proof. Set Z U (z) = u1 (z) − u2 (z) =

[b1 (ζ, u1 (ζ)) − b2 (ζ, u2 (ζ))] dζ. [0,z] 2

Using assertions (j) − (jjj), we get, for each z ∈ [0, T ] R U (z) ≤ [0,z] [b2 (ζ, u1 (ζ)) − b2 (ζ, u2 (ζ))] dζ R ≤ [0,z] [b2 (ζ, u1 (ζ)) − b2 (ζ, u2 (ζ))] 1[0,+∞) (U (ζ)) dζ R ≤ [0,z] ρ (U (ζ)) 1(−∞,0] (U (ζ)) dζ R = [0,z] ρ (U + (ζ)) dζ. This implies that

Z

¡ ¢ ρ U + (ζ) dζ,

+

U (z) ≤ [0,z]

2

and by Bihari’s Lemma for integrals in the plane, U + (z) = 0 on [0, T ] . We can also see that the above comparison theorem remains true under the following assumption  2  (j) For each z ∈ [0, T ] , f (z, x) is nondecreasing function of x (C) (jj) |f (z, x) − f (z, y)| ≤ ρ (|x − y|) , for all z ∈ [0, T ]2  (jjj) b1 (z, x) ≤ f (z, x) ≤ b2 (z, x) where ρ is as before.

5

Existence of a strong solution

The purpose of this section is to discuss the existence of a strong solution of Equation (11). For this we will make use of an approach similar to that con1 sidered by [16] in the case H = H 0 = . In fact, there is no stochastic integral 2 in the Equation (11) and in the proof of the existence we just use the fact that the fractional Bownian sheet is a Gaussian process. Let us first recall ¡ ¢some definitions and preliminaries results. For d ≥ 1, let us denote by B Rd the Borel σ-algebra on Rd and by λm the Lebesge measure on Rm . ¡ ¢ Definition 7 Let M be the set of measures on B Rd . We say¡ that ¢ M is equiabsolutely continuous with respect to a given measure ν on B Rd if for every ¡ ¢ ε > 0 there exists a δ > 0 such that µ (A) < ε for all µ ∈ M and all A ∈ B Rd satisfying ν (A) < δ.

10

Definition 8 Let D be a domain in Rd and let X = {Xz , z ∈ D} be a measurable random field taking values in Rm . The measure defined on B (Rm ) by µZ ¶ µX (A) = E 1A (Xz ) dz , A ∈ B (Rm ) D

is called the occupation measure of X. We next state a lemma whose proof is given by Gy¨ongy in [7]. Let Xn = {Xn (z) ; z ∈ D, n ≥ 1} be a sequence of Rm -valued random fields on a bounded domain D ⊂ Rd and let {fn , n ≥ 1} be a sequence of uniformly bounded functions fn : Rm → R such that the following conditions are satisfied: (a) Xn (z) converges in probability to a random variable X (z) for every z ∈ D. (b) The occupation measures µXn of Xn form an equi-absolutely continuous set of measures with respect to λm . (c) The functions fn converge to f in the measure λm on compact sets, i.e. for every ε > 0 and R > 0, lim λm {x ∈ Rm : |x| < R, |fn (x) − f (x)| ≥ ε} = 0.

n→∞

We then have the following result: Lemma 9 Suppose that Xn , X, fn and f verify (a)-(c). Then for every p ≥ 1 µZ ¶ p lim E |fn (Xn (z)) − f (X (z))| dz = 0 n→∞

H,H

0

D H,H 0

Let B = {B , z ∈ [0, T ]2 } be a fractional Brownian sheet with ¡ ¢2 0 Hurst parameter (H, H ) ∈ 0, 12 . Fix M > 0 and denote by SM the set of all 2 continuous process y on [0, T ] of the form Z 0 y(z) = x + b(ζ)dζ + BzH,H [0,z]

where b(.) is an FzB -adapted process bounded by M and x ∈ R. Let also SM 2 be the set of continuous processes y on [0, T ] such that there exists a sequence of processes (yn )n∈N of SM with lim yn (z, ω) = y (z, ω)

n→∞

for almost all (z, ω) ∈ [0, T ]2 × Ω. We will need the following lemma. Lemma n 10 Let {un , n ≥ 1} be a family of processes oof SM , u a process of 2 SM and fn (z, x) , f (z, x) ; n ≥ 1, z ∈ [0, T ] , x ∈ R a family of functions verifying: 2

(i) un (z) converges to u (z) in probability for every z ∈ [0, T ] (ii) fn is bounded uniformly in n by a fixed constant M 2 (iii) fn converges to f in the Lebesgue measure on the compact sets of [0, T ] × R. 11

Then for every p ≥ 1, ÃZ

! p

lim E

n→∞

[0,T ]2

|fn (z, un (z)) − f (z, u (z))| dz

= 0.

Proof. Conditions (a) and (c) of Lemma 9 are verified for Xn (z) = (z, un (z)) and X (z) = (z, u (z)), and we have only to check condition (b). Suppose that 2 for any z ∈ [0, T ] Z 0 un (z) = x + bn (ζ)dζ + BzH,H [0,z]

where bn (.) is an FzB -adapted process bounded by M . Set, for n ≥ 1, µ ³ ³R ´´ R −1 b (ζ)dζ dWζ ρn = exp − [0,T ]2 KH,H 0 n [0,·] ζ ¶ ³ ³ ´´ 2 R R −1 b (ζ)dζ dζ − 12 [0,T ]2 KH,H 0 [0,·] n dPen

ζ

= ρn dP.

1 and using that 2 ¯ ´¯ ³R ¯ −1 ¯ ¯KH,H 0 [0,·] bn (ζ)dζ ¯ = ≤

Since H, H 0 ≤

bn (.) are bounded by M , we get ¯ ¯ ¯ ¯ H− 12 H 0 − 21 12 −H, 12 −H 0 21 −H 12 −H 0 t I s t bn (ζ)¯ ¯s M cH,H 0 ,

where cH,H 0 is the constant appearing in (12). It follows that for any α ≥ 1 µ ´´ ³ ³R R −1 −α dWζ E (ρn ) = exp α [0,T ]2 KH,H b (ζ)dζ 0 n [0,·] ζ¶ ´´ ³ ³ 2 R R −1 + α2 [0,T ]2 KH,H b (ζ)dζ dζ 0 [0,·] n ζ

≤ Cα where Cα is a constant independent of n. From Girsanov theorem in the plane e e (see ³ [6]), Pn0 ´is a probability measure, and under Pn , un (.) has the same law as H,H e under P . Let us denote by En the expectation with respect to Pen x + B· ³ ´ ¡ ¢ 0 and by Y the process z, x + BzH,H . If A ∈ B R3 , by Schwartz inequality we get ³R ´ ³ ´ R e n ρ−1 E [0,T ]2 1A (ζ, un (ζ)) dζ = E 1 (ζ, u (ζ)) dζ 2 A n n [0,T ] h ¡ i1/2 · ³R ´2 ¸1/2 ¢ −2 e e ≤ En ρn En [0,T ]2 1A (ζ, un (ζ)) dζ h ³R ´i1/2 en ≤ CT E 2 1A (ζ, un (ζ)) dζ [0,T ] ≤ CT (µY (A)) 12

1/2

³ ´ 0 which ends the proof, since the law of z, x + BzH,H is equivalent to λ3 . Now we are able to prove the existence and uniqueness result. 2

Theorem 11 Suppose that b : [0, T ] ×R → R is a measurable function verifying (H1 )

sup z∈[0,T ]2

sup |b (z, r)| ≤ M < ∞ r∈R 2

(H2 ) ∀ z ∈ [0, T ] , b (z, r) is a nondecreasing function of r. Then there exists a unique FzB -adapted solution to Equation (11). To prove the theorem we need the following lemma (see Lemma 2.4 in [15]) which provides an increasing approximation of function satisfying assumptions (H1 ) and (H2 ) . 2

Lemma 12 Let b : [0, T ] × R → R is a measurable function verifying (H1 ) and (H2 ) . Then there exists a family of measurable functions n o 2 bn (z, x) ; n ≥ 1, z ∈ [0, T ] , x ∈ R such that  · lim bn (z, x) = b (z, x) , dz ⊗ dx a.e.   n→∞   2    · x 7→ bn (z, x) is nondecreasing, for all n ≥ 1, z ∈ [0, T ] 2 · n 7→ bn (z, x) is nondecreasing, for all x ∈ R, z ∈ [0, T ]   · bn (z, .) is Lipschitz uniformly in z, ∀n ≥ 1, with constant Ln     sup sup |b (z, r)| ≤ M.  · sup n≥1

z∈[0,T ]2

r∈R

n o 2 Proof. (Existence) We pick a family bn (z, x) ; n ≥ 1, z ∈ [0, T ] , x ∈ R as in Lemma 12. Then for any n ≥ 1, Equation (11) has a unique solution which we denote by un . It follows from Theorem 6 that un is nondecreasing and bounded in n, and therefore it has a limit. We set lim un (z) = u (z) .

n→∞

By Lemma 10, we have

Z

L1 − lim

n→∞

Z bn (ζ, un (ζ)) dζ =

[0,z]

b (ζ, u(ζ)) dζ, [0,z]

which allows us to show that u satisfies Equation (11). (Uniqueness) Let v be another solution. Then for any n ≥ 1, v−un verifies, R v(z) − un (z) = [0,z] [b(ζ, v (ζ)) − bn (ζ, un (ζ))] dζ R ≥ [0,z] [bn (ζ, u1 (ζ)) − bn (ζ, u2 (ζ))] dζ R − ≥ −Ln [0,z] (v(ζ) − un (ζ)) dζ. We deduce that v(z) ≥ un (z) a.s., and by a limit argument we show that v(z) ≥ u(z) a.s. On the other hand the laws of v and u are the same, so we obtain v = u a.s. 13

6

Euler’s approximation scheme

In this section we study the convergence of Euler’s scheme for Equation (11). Throughout this section Cp denotes a constant which changes from line to line. To simplify the notation assume that T = 1. Inspired by [16], we define an approximation scheme for the solution of equation (11) as follows. For any n ≥ 1 and 0 ≤ i, j ≤ n we set ¡ ¢ i j n , tnj = , zi,j = sni , tnj n n £ ¤ n n n We also set Ii,j = zi,j , zi+1,j+1 if 0 ≤ i, j ≤ n − 1. For every n ≥ 1, we define the random field un by  un (0, t) = un (s, 0) = x      ³ ´  ¡ n ¡ n ¢¢   1 H,H 0  n n (un ) + 4 B , 4 = 2 bn zi,j , un zi,j I I  n i,j i,j         if 0 ≤ i, j ≤ n − 1     ¡ n ¢¢ ¡ ¡ n ¢ ¡ n ¢ (16) (s − sni ) − un zi,j + n un zi+1,j un (z) = un zi,j     ¡ n ¢¢ ¡ ¡ n ¢    (t − tni ) − un zi,j +n un zi,j+1        n n  n (un ) (s − s ) (t − t ) , +n2 4Ii,j  i i       n if 0 ≤ i, j ≤ n − 1, z ∈ Ii,j . ³ ´ 0 n We will refer to this scheme as Eq bn , B H,H . Note that for z ∈ Ii,j , un (z) is obtained by Euler’s polygonal approximation. We assume the following assumptions: sni =

(H3 ) (H4 )

sup z∈[0,T ]2

lim

n→∞

sup |bn (z, r)| ≤ M. r∈R

sup

|bn (z, r) − b (z, r)| = 0

z∈[0,T ]2

dr − a.e.

We will show that the sequence un (., x) converge to u (., x) in Lp , p ≥ 2, uni2 formly in z ∈ [0, 1] and x ∈ B where B is a compact subset of R. This is given in the following theorem. Theorem 13 Suppose that b(z,³x) satisfies´ the assumptions of Theorem 11. 0 Then the process un given by Eq bn , B H,H verifies à lim

n→∞

sup E x∈B

! p

sup |un (z, x) − u(z, x)|

z∈[0,1]2

14

= 0.

We start with the following a priori estimates of un and u which we obtain using assumptions H1 and H3 . Lemma 14 Assume that assumptions H1 and H3 hold. Then for every p > 2 there exists a constant Cp > 0 such that à ! (i) sup sup E x∈B n≥1 à (ii) sup E x∈B

p

sup |un (t, x)| !

z∈[0,1]2

sup |u(t, x)| z∈[0,1]

2

p

≤ Cp .

≤ Cp .

For the following lemmas, we will need some additional notations. If z = 2 (s, t) ∈ [0, 1] and n ≥ 1, we will set µ ¶ [ns] [nt] k n (z) = , . n n 2

We also set, for a process X in [0, 1] and a measurable function f : [0, 1]×R → R f (X, k n ) (z) = f (k n (z) , Xk n (z) ). Finally, we will denote by λ the Lebesgue measure in the plane. To be able to apply the Skorohod embedding theorem and to prove the convergence of the Euler’s approximation³ scheme, ´ we want to show the tightness of the fam2 ily {un (·, x), n ≥ 1} in C [0, 1] . This is the reason for giving the following lemma. Lemma 15 Assume that assumptions H1 and H3 hold. Then for every p > 2 there exists a constant Cp > 0 such that ¯p ¢ ¡¯ p H∧H 0 ) (i) sup sup E ¯4[z1 ,z2 ] un (z, x)¯ ≤ Cp (λ ([z1 , z2 ])) ( . x∈B n≥1 ¯p ¢ ¡¯ p H∧H 0 ) (ii) sup E ¯4[z1 ,z2 ] u(z, x)¯ ≤ Cp (λ ([z1 , z2 ])) ( . x∈B

n n , z2 = zk,l with i < k and j < l. We have Proof. Suppose first that z1 = zi,j Z 0 n un (zi,j , x) = x + BzH,H + bn (un , k n ) (ζ)dζ, n i,j n [0,zi,j ]

and

Z 0

4[zn , zn ] un = 4[zn , zn ] B H,H + i,j i,j k,l k,l

n , zn [zi,j k,l ]

bn (un , k n ) (ζ)dζ.

It follows that E

¡ ¯ ¯4[z

1 ,z2 ]

¯p ¢ p H∧H 0 ) (un )¯ ≤ Cp (λ ([z1 , z2 ])) ( .

The cases where z1 and z2 are not of this form are treated as in [15]. The next two lemmas provided the main ingredients to show that the limit of un (·) satisfies the Equation (11). 15

n o 2 Lemma 16 Let un (z) ; n ≥ 1, z ∈ [0, 1] be the sequence of processes constructed by (16). Then, for any p > 1 and 0 < ε < 1, there exists Nε and Kp,ε 2 such that for any measurable bounded function h : [0, 1] × R → R and n > Nε "Z # p

E [ε,1]2

|h (z, un (kn (z)))| dz ≤ Kp,ε khkp .

n o 2 Moreover, if there exists a subsequence unk (z) ; k ≥ 1, z ∈ [0, 1] converging dP ⊗ dz a.s. to a process u, the inequality still holds for u. 2

Proof. For any z ∈ [0, 1] , we have Z 0 un (k n (z)) = x + bn (un , k n ) (ζ) dζ + BkH,H n (z) [0,k n (z)]

Set, for n ≥ 1, µ ρn

=

exp − R 1

−2 dPen

=

³

R [0,1]2

[0,1]2

³

−1 KH,H 0 ³R

−1 KH,H 0

³R

´´ b (u , k n )(ζ)dζ [0,·] n n

b (u , k n )(ζ)dζ [0,·] n n

´´2 ζ

dWζ ζ ¶ dζ

ρn dP

1 and using that bn (.) is bounded by M , we get, as in the proof 2 of Lemma 10, the estimate E (ρ−α n ) ≤ Cα , where Cα is a constant independent of n. From Girsanov theorem in the plane (see [6]), Pen is a probability measure, ³ ´ H,H 0 e and under Pn , un (k n (z)) has the same law as the process x + Bk n (z) under µ ³ ´2H ³ ´2H 0 ¶ [nt] . We denote P . In particular un (kn (z)) has the law N x, [ns] n n Since H, H 0 ≤

by p (r, z) the probability density of un (kn (z)) under Pen . ³Let us denote ´ by H,H 0 e e En the expectation with respect to Pn and by Y the process x + Bz . By H¨older’s inequality hR i h i R p p e n ρ−1 E [ε,1]2 |h (ζ, un (k n (ζ)))| dζ = E |h (ζ, u (k (ζ)))| dζ 2 n n n [ε,1] h i1/q h R i1/p p −q e n (ρ ) en ≤ C E E |h (ζ, un (k n (ζ)))| dζ n [ε,1]2 h R i1/p p en |h (ζ, un (k n (ζ)))| dζ ≤ C E [ε,1]2 ³R ´1/p R p ≤ C [ε,1]2 R |h (ζ, r)| p (k n (ζ), r) drdζ Ã ! ≤

C

sup z∈[ε,1]2

µ ¶1/(H∧H 0 ) 2 if n ≥ . ε 16

sup p (z, r) khkp r∈R

n o 2 Lemma 17 Let un (z) ; n ≥ 1, z ∈ [0, 1] be the sequence of processes constructed by (16) and {um (z) ; m ∈ J} a subsequence converging dP ⊗ dz a.s. to a process u . Then, for any 0 < ε < 1, p > 1 "Z # lim

m→∞, m∈J

p

E [ε,1]2

|bm (um , km ) (z) − b (z, u (z) )| dz = 0.

The proof of this lemma is analogous to that of Lemma 3.6 in [15]. Now we are ready to prove the convergence of Euler’s approximation scheme. Proof of Theorem 13. Assume that there exist δ > 0, x ∈ R and a sequence of natural numbers (n(j))j≥0 and a sequence of real numbers and ¡ ¢ xn(j) j≥0 converging to x such that à δ ≤inf E j

¯ ¯p sup ¯un(j) (z, xn(j) ) − u(z, x)¯

! .

t∈[0,1]

We claim that this assumptions leads to a contradiction with the hypotheses of the Theorem. To simplify the notation set un = u³n(j) and xn = xn(j) . From ´ 0

Lemma 15 it follows that the law of the process un (., xn ), u(., xn ), B H,H ³ ´⊗3 2 is tight on C [0, 1] , where u(., xn ) is the solution of (11) and un (., xn ) is constructed by (16) with the initial data xn . By the µ Skorohod theorem there ex¶ ´ ³ ^ H,H 0 e F, e Pe and a sequence u f (., x ), ve (., x ), B ists a probability space Ω, n

³

n

0

´

n

n

n

with the same distribution as un (., xn ), u(., xn ), B H,H which converges al³ ´ ³ ´⊗3 2 ^ H,H 0 most surely to u e (., x) , ve (., x) , B in C [0, 1] . Moreover, for every 2

n z ∈ [ε, 1] with the notation k n (z) = zi,j , we can see that

u fn (z, xn ) =

where

R ^ H,H 0 xn + [ε,z] bn (k n , u (k n (z)) fn ) (ζ) dζ + B R Rn − [k n (z),z] bn (k n , u fn ) (ζ) dζ + Cε bn (k n , u fn ) (ζ) dζ +A1

A1 (z)

¡ ¡ n ¢ ¡ n ¢¢ (s − sni ) = n u fn zi+1,j − u fn zi,j ¡ ¡ n ¢ ¡ n ¢¢ +n u fn zi,j+1 − u fn zi,j (t − tni ) n (f +n2 4Ii,j un ) (s − sni ) (t − tni ) ,

and Cε = [0, 1]2 \(ε, 1]2 . On the other hand, Z ^ H,H 0 ven (z, xn ) = xn + b (ζ, ven (ζ, xn ) ) dζ + B . z [0,z]

17

Applying Lemma 15 and assumption (H3 ), we deduce the following estimates 3Cp E (|A1 (z)|) ≤ H∧H 0 n ¯´ M ³¯R ¯ ¯ E ¯ [k n (z),z] bn (k n , u fn ) (ζ) dζ ¯ ≤ 2 ¯´ n ³¯R ¯ ¯ E ¯ Cε bn (k n , u fn ) (ζ) dζ ¯ ≤ 2M ε. Letting n tend to infinity we can easily see that u e³(., x) and´ve (., x) are two solue F, e Pe with the fractional tions of Equation (11) on the probability space Ω, ^ H,H 0 and the same initial data. On the other hand, using the Brownian sheet B statement (i) of Lemma 14 we have ! à 0

< δ≤E

sup |e u(t, x) − ve(t, x)|

p

z∈[0,1]2

Ã

= lim inf E n

!

sup |f un (t, xn ) − ven (t, xn )|

p

.

z∈[0,1]2

We may notice finally that, by the pathwise uniqueness, we have u e (., x) = ve (., x) almost surely. That is a contradiction. Acknowledgement 18 This work was completed during a stay of Youssef Ouknine at the IMUB (Institut de Matem` atica de la Universitat de Barcelona). He would like to thank the IMUB for hospitality and support.

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