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Vol. 14 (2009), Paper no. 18, pages 477–499. Journal URL http://www.math.washington.edu/~ejpecp/
Homogenization of semilinear PDEs with discontinuous averaged coefficients K. Bahlali∗†
A. Elouaflin‡ §
E. Pardoux ¶
Abstract We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coefficients have averages in the Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic differential equation (BSDE) in the Jakubowski S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "L p −viscosity solution" introduced in [7]. The existence of L p −viscosity solution to the averaged PDE is proved here by using BSDEs techniques. Key words: Backward stochastic differential equations (BSDEs), L p -viscosity solution for PDEs, homogenization, Jakubowski S-topology, limit in the Cesaro sense. AMS 2000 Subject Classification: Primary 60H20, 60H30, 35K60. Submitted to EJP on May 13, 2008, final version accepted January 28, 2009.
∗
Partially supported by PHC Tassili 07MDU705 and Marie Curie ITN, no. 213841-2 IMATH, UFR Sciences, USVT, B.P. 132, 83957 La Garde Cedex, France. e-mail:
[email protected] ‡ UFRMI, Université de Cocody, 22 BP 582 Abidjan, Côte d’Ivoire e-mail:
[email protected] § Supported by AUF bourse post-doctorale 07-08, Réf.: PC-420/2460. ¶ LATP, CMI Université de Provence, 39 rue Joliot-Curie,13453 Marseille. e-mail:
[email protected]
†
477
1 Introduction Homogenization of a partial differential equation (PDE) is the process of replacing rapidly varying coefficients by new ones such that the solutions are close. Let for example a be a one dimensional periodic function which is positive and bounded away from zero. For ǫ > 0, we consider the operator x Lǫ = d iv(a( )∇) ǫ For small ǫ, Lǫ can be replaced by L = d iv(a∇) where a is the averaged (or limit, or effective) coefficient associated to a. As ǫ is small, the solution of the parabolic equation ∂ t u = Lǫ u, u(0, x) = f (x) is close to the corresponding solution with Lǫ replaced by L. The probabilistic approach to homogenization is one way to prove such results in the periodic or ergodic case. It is based on the asymptotic analysis of the diffusion process associated to the operator Lǫ . The averaged coefficient a is then determined as a certain "mean" of a with respect to the invariant probability measure of the diffusion process associated to L. There is a vast literature on the homogenization of PDEs with periodic coefficients, see for example the monographs [3; 12; 21] and the references therein. There also exists a considerable literature on the study of asymptotic analysis of stochastic differential equations (SDEs) with periodic structures and its connection with homogenization of second order partial differential equations (PDEs). In view of the connection between BSDEs and semilinear PDEs, this probabilistic tool has been used in order to prove homogenization results for certain classes of nonlinear PDEs, see in particular [4; 5; 6; 9; 11; 13; 19; 23; 24] and the references therein. The two classical situations which have been mainly studied are the cases of deterministic periodic and random stationary coefficients. This paper is concerned with a different situation, building upon earlier results of Khasminskii and Krylov. In [15], Khasminskii & Krylov consider the averaging of the following family of diffusions process Z t x1 1 1, ǫ Ut = + ϕ(Us1, ǫ , Us2, ǫ )dWs , ǫ ǫ 0 (1.1) Z t Z t 2, ǫ U = x + fs , b(1) (Us1, ǫ , Us2, ǫ )ds + σ(1) (Us1, ǫ , Us2, ǫ )d W 2 t 0
0
1, ǫ Ut
2, ǫ
where for each ǫ > 0 small, is a one-dimensional null-recurrent fast component and U t is a d–dimensional slow component. The function ϕ (resp. σ(1) , resp. b(1) ) is IR-valued (resp. IRd×(k−1) f) is a k-dimensional standard Brownian motion whose component valued, resp. IRd -valued). (W, W f W (resp. W ) is one dimensional (resp. (k-1)-dimensional). Define now (X 1,ǫ , X 2,ǫ ) = (ǫU 1,ǫ , U 2,ǫ ). 1,ǫ 2,ǫ The process {X tǫ := (X t , X t ), t ≥ 0} solves the SDE Z t 1, ǫ Xs 1, ǫ 2, ǫ dWs , , Xs X = x1 + ϕ t ǫ 0 (1.2) 1, ǫ 1, ǫ Z t Z t Xs Xs 2, ǫ 2, ǫ (1) (1) X 2, ǫ = x + fs , , Xs , Xs ds + dW σ b 2 t ǫ ǫ 0 0 478
They define the averaged coefficients as limits in the Cesaro sense. With the additional assumption 1, ǫ 2, ǫ that the presumed SDE limit is weakly unique, they prove that the process (X t , X t ) converges in distribution towards a Markov diffusion (X t1 , X t2 ). As a byproduct, they derive the limit behavior 1, ǫ 2, ǫ of the linear PDE associated to (X t , X t ), in the case where weak uniqueness of the limiting PDE 1,2 holds in the Sobolev space Wd+1,loc (IR+ × IRd ) of all funcions u(t, x) defined on IR+ × IRd such that (IR+ × IRd ). both u and all the generalized derivatives D t u, D x u, and D2x x u belong to L ld+1 oc
In the present note, we extend the results of [15] to parabolic semilinear PDEs. Note that the limiting coefficients can be discontinuous. More precisely, we consider the following sequence of semi-linear PDEs, indexed by ǫ > 0, ǫ ∂ v (t, x , x ) = (L ǫ v ǫ )(t, x , x ) + f ( x 1 , x , v ǫ (t, x , x )), t > 0 1 2 1 2 2 1 2 ∂t ǫ (1.3) v ǫ (0, x 1 , x 2 ) = H(x 1 , x 2 ); (x 1 , x 2 ) ∈ IR × IRd . L ǫ := a00 (
where ϕ, σ
(1)
and b
(1)
x1 ǫ
, x2)
∂2 ∂ 2 x1
+
d X
ai j (
i, j=1
x1 ǫ
, x2)
∂2 ∂ x 2i ∂ x 2 j
+
d X
(1)
bi (
i=1
x1 ǫ
, x2)
∂ ∂ x 2i
,
are those defined above in equation (1.1), a00 :=
1 2
ϕ2,
ai j :=
1 2
(σ(1) σ(1) ∗ )i j , i, j = 1, ..., d,
and the real valued measurable functions f and H are defined on IRd+1 × IR and IRd+1 respectively.
We put
b :=
0 b(1)
,
a(x) :=
We write
B :=
W f W
1 2
∗
(σσ )(x),
with σ :=
and
ǫ
X :=
X 1, ǫ X 2,ǫ
ϕ 0 0 σ(1)
.
.
The PDE (1.3) is then connected to the system of SDE – BSDE Zs Zs X r1, ǫ 2, ǫ X 1, ǫ ǫ Xs = x + b( , X r )d r + σ( r , X r2, ǫ )d B r , ǫ ǫ 0 0 Z t Z t 1, ǫ Xr ǫ 2, ǫ ǫ Y ǫ = H(X ǫ ) + , X r , Yr )d r − Z rǫ d M rX , ∀ s ∈ [0, t] f( s t ǫ s s
(1.4)
ǫ
where M X is the martingale part of the process X ǫ i. e. Zs X 1, ǫ Xǫ σ( r , X r2, ǫ )d B r , Ms = ǫ 0
0 ≤ s ≤ t.
Note that Y0ǫ does depend upon the pair (t, x) where x is the initial condition of the forward SDE part of (1.4), and t is the final time of the BSDE part of (1.4). It follows from e. g. Remark 2.6 in [22] that under suitable conditions upon the coefficients {v ǫ (t, x) := Y0ǫ , t ≥ 0, x = (x 1 , x 2 ) ∈ IRd+1 } solves the PDE (1.3). The aim of the present paper is 479
Rt ǫ 1. to show that for each t > 0, x ∈ IRd+1 , the sequence of processes (X sǫ , Ysǫ , s Z rǫ d M rX )0≤s≤t Rt converges in law to the process (X s , Ys , s Z r d M rX )0≤s≤t which is the unique solution to the system of SDE – BSDE Zs Zs ¯b(X r )d r + ¯ r )d B r , 0 ≤ s ≤ t. X =x+ σ(X s 0 0 (1.5) Z t Z t X Y = H(X ) + ¯ f (X r , Yr )d r − Z r d M , 0 ≤ s ≤ t, s t r
s
s
¯ ¯b and f¯ are respectively the average of σ, b and where M X is the martingale part of X and σ, f , in a sense which will be made precise below; 2. deduce from the first result that for each (t, x), v ǫ (t, x 1 , x 2 ) −→ v(t, x 1 , x 2 ), where v solves the following averaged PDE in the L p -viscosity sense ∂ v (t, x , x ) = ( ¯L v)(t, x , x ) + f¯(x , x , v(t, x , x )) t > 0, 1 2 1 2 1 2 1 2 (1.6) ∂t v(0, x 1 , x 2 ) = H(x 1 , x 2 ), with ¯L =
X i, j
a¯i j (x 1 , x 2 )
∂2 ∂ xi∂ x j
+
X i
¯bi (x 1 , x 2 )
∂ ∂ xi
the averaged operator. The method used to derive the averaged BSDE is based on weak convergence in the S-topology and is close to that used in [23] and [24]. In our framework, we show that the limiting system of SDE – BSDE (1.5) has a unique solution. However, due to the discontinuity of the coefficients, the classical viscosity solution is not defined for the averaged PDE (1.6). We then use the notion of "L p −viscosity solution". We use BSDE techniques to establish the existence of L p −viscosity solution for the averaged PDE. The notion of L p -viscosity solution has been introduced by Caffarelli et al. in [7] to study fully nonlinear PDEs with measurable coefficients. Note however that although the notion of a L p -viscosity solution is available for PDEs with merely measurable coefficients, continuity of the solution is required. In our situation, the lack of L 2 -continuity property for the flow X x := (X 1, x , X 2, x ) transfers the difficulty to the backward one and hence we cannot prove the L 2 continuity of the process Y . To overcome this difficulty, we establish weak continuity for the flow x 7→ (X 1, x , X 2, x ) and using the fact that Y0x is deterministic, we derive the continuity property for Y0x . The paper is organized as follows: In section 2, we make precise some notations and formulate our assumptions. Our main results are stated in section 3. Section 4 and 5 are devoted to the proofs.
480
2 Notations and assumptions 2.1 Notations For a given function g(x 1 , x 2 ), we define 1
+
g (x 2 ) := lim
x 1 →+∞
−
g (x 2 ) := lim
x1 1
x 1 →−∞
Z
x1
g(t, x 2 )d t Z
x1
0 x1
g(t, x 2 )d t 0
The average, in Cesaro sense, of g is defined by g ± (x 1 , x 2 ) := g + (x 2 )1{x 1 >0} + g − (x 2 )1{x 1 ≤0} Let ρ(x 1 , x 2 ) := a00 (x 1 , x 2 )−1 (= [ 21 ϕ 2 (x 1 , x 2 )]−1 ) and denote by ¯b(x 1 , x 2 ), a¯(x 1 , x 2 ) and f¯(x 1 , x 2 , y), the averaged coefficients defined by ¯bi (x 1 , x 2 ) = a¯i j (x 1 , x 2 ) = f¯(x 1 , x 2 , y) =
(ρ bi )± (x 1 , x 2 )
i = 1, ..., d
, ρ ± (x 1 , x 2 ) (ρai j )± (x 1 , x 2 )
, ρ ± (x 1 , x 2 ) (ρ f )± (x 1 , x 2 , y) ρ ± (x 1 , x 2 )
i, j = 0, 1, ..., d .
1
¯ 1 , x 2 ) = (¯ σ(x a(x 1 , x 2 )) 2
where a¯(x 1 , x 2 ) denotes the matrix (¯ ai j (x 1 , x 2 ))i, j . It is worth noting that ¯b, a¯ and f¯ may be discontinuous at x 1 = 0.
2.2 Assumptions. We consider the following conditions. (A1) The functions b(1) , σ(1) , ϕ are uniformly Lipschitz in the variables (x 1 , x 2 ). (A2) For each x 1 , the first and second order derivatives with respect to x 2 of these functions are bounded continuous functions of x 2 . (A3) a(1) := 12 (σ(1) σ(1) ∗ ) is uniformly elliptic, i. e. ∃Λ > 0; Moreover, there exist positive constants C1 , C2 , C3 such that ( (i) C1 ≤ a00 (x 1 , x 2 ) ≤ C2 (ii)
∀x, ξ ∈ IRd ,
|a(1) (x 1 , x 2 )| + |b(x 1 , x 2 )|2 ≤ C3 (1 + |x 2 |2 ).
481
ξ∗ a(1) (x)ξ ≥ Λ|ξ|2 .
(B1) Let D x 2 ρ and D2x ρ denote respectively the gradient vector and the matrix of second derivatives 2 of ρ with respect to x 2 . We assume that uniformly with respect to x 2 Z x1 1 ρ(t, x 2 )d t −→ ρ ± (x 2 ) as x 1 → ±∞, x1 0 Z x1 1 D x 2 ρ(t, x 2 )d t −→ D x 2 ρ ± (x 2 ) as x 1 → ±∞, x1 0 Z x1 1 D2x ρ(t, x 2 )d t −→ D2x ρ ± (x 2 ) as x 1 → ±∞. 2 2 x1 0 (B2) For every i and j, the coefficients ρ bi , D x 2 (ρ bi ), D2x (ρ bi ), ρai j , D x 2 (ρai j ), 2
D2x (ρai j ) have averages in the Cesaro sense. 2
(B3) For every function k ∈ {ρ bi , D x 2 (ρ bi ), D2x (ρ bi ), ρai j , D x 2 (ρai j ), D2x (ρai j )}, there 2
2
exists a bounded function α : IRd+1 → IR such that Z x1 1 k(t, x 2 )d t − k± (x 1 , x 2 ) = (1 + |x 2 |2 )α(x 1 , x 2 ), x 1
0
(2.1)
sup |α(x 1 , x 2 )| = 0. |x 1 |−→∞ d x 2 ∈IR lim
(C1) (i) The coefficient f is uniformly Lipschitz in (x 1 , x 2 , y) and, for each x 1 ∈ IR, its derivatives in (x 2 , y) up to and including second order derivatives are bounded continuous functions of (x 2 , y). (ii) There exists positive constant K such that for every (x 1 , x 2 , y),
| f (x 1 , x 2 , y)| ≤ K(1 + |x 2 | + | y|).
(iii) H is continuous and bounded. (C2) ρ f has a limit in the Cesaro sense and there exists a bounded measurable function β : IRd+2 → IR such that Z x1 1 ρ(t, x 2 ) f (t, x 2 , y)d t − (ρ f )± (x 1 , x 2 , y) = (1 + |x 2 |2 + | y|2 )β(x 1 , x 2 , y) x 1 0 (2.2) lim sup |β(x , x , y)| = 0, |x 1 |→∞
1
2
d
(x 2 , y)∈IR ×IR
(C3) For each x 1 , ρ f has derivatives up to second order in (x 2 , y) and these derivatives are bounded and satisfy (C2). Throughout the paper, (A) stands for conditions (A1), (A2), (A3); (B) for conditions (B1), (B2), (B3) and (C) for (C1), (C2), (C3).
482
3 The main results Consider the equation
Z X tx
Z
t
t
¯b(X x )ds + s
=x+ 0
0
¯ sx )d Bs , t ≥ 0. σ(X
(3.1)
Assume that (A), (B) hold. Then, from Khasminskii & Krylov [15] and Krylov [18], we deduce that for each fixed, x ∈ IRd+1 the process X ǫ := (X 1, ǫ , X 2, ǫ ) converges in distribution to the process X := (X 1 , X 2 ) which is the unique weak solution to SDE (3.1). We now define the notion of L p -viscosity solution of a parabolic PDE. This notion has been introduced by Caffarelli et al. in [7] to study PDEs with measurable coefficients. Presentations of this topic can be found in [7; 8]. Let g : IRd+1 × IR 7−→ IR be a measurable function and ¯L :=
X i, j
a¯i j (x)
∂2 ∂ xi∂ x j
+
X i
¯bi (x)
∂ ∂ xi
denote the second order PDE operator associated to the SDE (3.1). We consider the parabolic equation ∂ v (t, x) = ( ¯L v)(t, x) + g(x, v(t, x)), t ≥ 0 ∂t v(0, x) = H(x).
(3.2)
Definition 3.1. Let p be an integer such that p > d + 2. (a) A function v ∈ C [0, T ] × IRd+1 , IR is a L p -viscosity sub-solution of the PDE (3.2), if for every 1, 2 x ∈ IRd+1 , v(0, x) ≤ H(x) and for every ϕ ∈ Wp, l oc IR+ × IRd+1 , IR and (bt , b x ) ∈ (0, T ] × IRd+1 at which v − ϕ has a local maximum, one has ¾ ½ ∂ϕ (t, x) − ( ¯L ϕ)(t, x) − g(x, v(t, x)) ≤ 0. ess lim inf ∂t (t, x)→(bt , b x) (b) A function v ∈ C [0, T ] × IRd+1 , IR is a L p -viscosity super-solution of the PDE (3.2), if for 1, 2 x ) ∈ (0, T ] × every x ∈ IRd+1 , v(0, x) ≥ H(x) and for every ϕ ∈ Wp, l oc IR+ × IRd+1 , IR and (bt , b
IRd+1 at which v − ϕ has a local minimum, one has ½ ¾ ∂ϕ ess lim sup (t, x) − ( ¯L ϕ)(t, x) − g(x, v(t, x)) ≥ 0. ∂t (t, x)→(bt , b x)
Here, G(t, x, ϕ(s, x)) is merely assumed to be measurable upon the variable x =: (x 1 , x 2 ). (c) A function v ∈ C [0, T ] × IRd+1 , IR is a L p -viscosity solution if it is both a L p -viscosity subsolution and super-solution.
483
x ) of Remark 3.2. Condition (a) means that for every ǫ > 0, r > 0, there exists a set A ⊂ B r (bt , b positive measure such that, for every (s, x) ∈ A, ∂ϕ ∂s
(s, x) − ( ¯L ϕ)(t, x) − g(x, v(t, x)) ≤ ǫ.
The main results are (the S–topology is explained in the Appendix below) Theorem 3.3. Assume (A), (B), (C) hold. Then, for any (t, x) ∈ IR+ × IRd+1 , there exists a process (X s , Ys , Zs )0≤s≤t such that, (i) the sequence of process X ǫ converges in law to the continuous process X, which is the unique weak solution to SDE (1.5), in C([0, t]; IRd+1 ) equipped with the uniform topology. Rt ǫ (ii) the sequence of processes (Ysǫ , s Z rǫ d M rX )0≤s≤t converges in law to the process Rt (Ys , s Z r d M rX )0≤s≤t in D([0, t]; IR2 ), where M X is the martingale part of X , equipped with the S– topology. (iii) (Y,Z) is the unique solution to BSDE (1.5) such that, Rt (a) (Y,Z) is F X −adapted and (Ys , s Z r d M rX )0≤s≤t is continuous. Rt (b) IE sup0≤s≤t |Ys |2 + 0 |Z r σ(X r )|2 d r < ∞ The uniqueness means that, if (Y 1 , Z 1 ) and (Y 2 , Z 2 ) are two solutions of BSDE (1.5) satisfying (iii) ¯2 ¯ ¯2 R t ¯ 1 2 1 2 ¯Y − Y ¯ + ¯ Z σ(X ) − Z σ(X )¯ d r = 0, i. e. since σσ∗ is elliptic (a)-(b) then, IE sup 0≤s≤t
(see (A3)),
Ys1
=
Ys2
s
s
0
∀0 ≤ s ≤ t, IP a. s., and
r
r
Zs1
=
r
r
Zs2
ds × dIP a. e.
Theorem 3.4. Assume (A), (B), (C) hold. For ǫ > 0, let v ǫ be the unique solution to the problem (1.3). Let (Ys(t,x) )s be the unique solution of the BSDE (1.5). Then (t,x)
(i) Equation (1.6) has a unique L p -viscosity solution v such that v(t, x) = Y0
.
(ii) For every (t, x) ∈ IR+ × IRd+1 , v ǫ (t, x) → v(t, x), as ǫ → 0.
4 Proof of Theorem 3.3. In all of this section, (t, x) ∈ IR+ × IRd+1 is arbitrarily fixed with t > 0.
Assertion (i) follows from [15] and [18]. Assertion (iii) can be established as in [23; 24]. We shall prove (ii). We first deduce from our assumptions (see in particular (A3) which says that the coefficients of the forward SDE part of (1.4) are bounded with respect to their first variable, and grow at most linearly in their second variable) Lemma 4.1. For all p ≥ 1, there exists constant C p such that for all ǫ > 0,
IE
sup 0≤s≤t
[|X s1,ǫ | p
+ |X s2,ǫ | p ]
484
≤ Cp .
4.1 Tightness and convergence for the BSDE. Proposition 4.2. There exists a positive constant C such that for all ǫ > 0 Z t ¯ ¯2 ¯ ¯2 ¯ Z ǫ σ(X ǫ )¯ d r ≤ C. IE sup ¯Y ǫ ¯ + s
0≤s≤t
r
r
0
Proof. We deduce from Itô’s formula (here and below X¯ r1, ǫ = X r1, ǫ /ǫ) Z t Z t Z t ¯ ¯2 ¯ Z ǫ σ(X ǫ )¯ d r) ≤ |H(X ǫ )|2 + K |Ysǫ |2 + |Yrǫ |2 d r + | f (X¯ r1, ǫ , X r2, ǫ , 0)|2 d r r r t 0
s
Z −2
s
t ǫ
s
〈Yrǫ , Z rǫ d MsX 〉.
It follows from well known results on BSDEs that we can take the expectation in the above identity (see e. g. [22]; note that introducing stopping times as usual and using Fatou’s Lemma would yield (4.1) below). We then deduce from Gronwall’s lemma that there exists a positive constant C which does not depend on ǫ, such that for every s ∈ [0, t], Z t IE |Y ǫ |2 ≤ CIE |H(X ǫ )|2 + | f (X¯ 1, ǫ , X 2, ǫ , 0)|2 d r t
s
r
r
0
and
Z
t
IE
¯ ¯ ¯ Z ǫ σ(X ǫ )¯2 d r r
r
0
Z |H(X tǫ )|2
≤ CIE
t
+ 0
|f
(X¯ r1, ǫ ,
X r2, ǫ ,
2
0)| d r
.
(4.1)
Combining the last two estimates and the Burkhölder-Davis-Gundy inequality, we get Z t Z t ¯ ¯ 1 ǫ 2 ǫ ǫ ¯2 ǫ 2 1, ǫ 2, ǫ 2 ¯ IE sup |Ys | + Z σ(X r ) d r ≤ CIE |H(X t )| + | f (X¯ r , X r , 0)| d r 2 0 r 0≤s≤t 0 In view of condition (C1) and Lemma 4.1, the proof is complete. We deduce immediately from Proposition 4.2 Corollary 4.3. sup |Y0ǫ | < ∞. ǫ>0
Proposition 4.4. For ǫ > 0, let Y ǫ be the process defined by equation (1.4) and M ǫ be its martingale part. The sequence (Y ǫ , M ǫ )ǫ>0 is tight in the space D ([0, t], IR) × D ([0, t], IR) endowed with the S-topology. Proof. Since M ǫ is a martingale, then by [20] or [14], the Meyer-Zheng tightness criteria is fulfilled whenever sup C V (Y ǫ ) + IE
sup |Ysǫ | + |Msǫ |
ǫ
< +∞.
0≤s≤t
where the conditional variation C V is defined in appendix A. >From [25], the conditional variation C V (Y ǫ ) satisfies Z t 1, ǫ 2, ǫ ǫ ǫ | f (X¯s , X s , Ys )|ds , C V (Y ) ≤ IE 0
Now clearly (4.2) follows from (C1), Lemma 4.1 and Proposition 4.2. 485
(4.2)
Proposition 4.5. There exists (Y, M ) and a countable subset D of [0, t] such that along a subsequence ǫn → 0, (i) (Y ǫn , M ǫn ) =⇒ (Y, M ) on D ([0, t], IR) × D ([0, t], IR) endowed with the S–topology. (ii) The finite dimensional distributions of Ys ǫn , Msǫn s∈D c converge to those of Ys , Ms s∈D c . (iii) (X 1,ǫn , X 2,ǫn , Y ǫn ) =⇒ (X 1 , X 2 , Y ) , in the sense of weak convergence in C([0, t], IRd+1 ) × D([0, t], IR), equipped with the product of the uniform convergence and the S topology. Proof. (i) From Proposition 4.4, the family (Y ǫ , M ǫ )ǫ is tight in D ([0, t], IR) × D ([0, t], IR) endowed with the S-topology. Hence along a subsequence (still denoted by ǫ), (Y ǫ , M ǫ )ǫ converges in law on D ([0, t], IR) × D ([0, t], IR) towards a càd-làg process (Y, M ). (ii) follows from Theorem 3.1 in Jakubowski [14].
(iii) According to Theorem 3.3 (i), (X 1,ǫ , X 2,ǫ ) =⇒ (X 1 , X 2 ) in C([0, t], IRd+1 ) equipped with the uniform topology. From assertion (i), (Y·ǫ )ǫ>0 is tight in D ([0, t], IR) equipped with the S–topology. Hence the subsequence ǫn can be chosen in such a way that (iii) holds.
4.2 Identification of the limit finite variation process. Proposition 4.6. Let (Y, M ) be any limit process as in Proposition 4.5. Then (i) for every s ∈ [0, t] \ D, Z t
Ys = H(X t ) + s
f¯(X r1 , X r2 , Y )d r − (M t − Ms ),
IE sup |Y |2 + |X 1 |2 + |X 2 |2 ≤ C; s s s
(4.3)
0≤s≤t
(ii) M is a Fs -martingale, where Fs := σ X r , Yr ,
0 ≤ r ≤ s augmented with the IP-null sets.
To prove this proposition, we need the following lemmas. Lemma 4.7. Assume (A), (B), (C2) and (C3). For x 2 ∈ IRd , y ∈ IR, let V ǫ (x 1 , x 2 , y) denote the solution of the following equation: x x a00 ( 1 , x 2 )D2 V ǫ (x 1 , x 2 , y) = f ( 1 , x 2 , y) − f¯(x 1 , x 2 , y), x 1 ∈ IR, x1 ǫ ǫ (4.4) ǫ V (0, x 2 , y) = D x 1 V ǫ (0, x 2 , y) = 0. Then, for some bounded functions β1 and β2 satisfying (2.2), x
(i) D x 1 V ǫ (x 1 , x 2 , y) = x 1 (1 + |x 2 |2 + | y|2 )β1 ( ǫ1 , x 2 , y), and the same is true with D x 1 V ǫ replaced by D x 1 D x 2 V ǫ and D x 1 D y V ǫ ; x
(ii) V ǫ (x 1 , x 2 , y) = x 12 (1 + |x 2 |2 + | y|2 )β2 ( ǫ1 , x 2 , y), and the same is true with V ǫ replaced by D x 2 V ǫ , D y V ǫ , D2x V ǫ , D2y V ǫ and D x 2 D y V ǫ . 2
486
Proof. We will adapt the idea of [15] to our situation. For ǫ > 0 and (z, x 2 , y) ∈ IRd+2 we set Z ǫz t t 1 ρ( , x 2 )g( , x 2 , y)d t Fǫ (z, x 2 , y) := ǫz 0 ǫ ǫ where g(z, x 2 , y) := f (z, x 2 , y) − f¯(ǫz, x 2 , y). We only treat the case where x 1 > 0. The same argument can be used in the case x 1 < 0. We successively use the definition of f¯ and assumptions (C2), to obtain Z x1 x1 t 1 t Fǫ ( , x 2 , y) = ρ( , x 2 ) f ( , x 2 , y)d t − (ρ f )+ (x 2 , y) ǫ x1 0 ǫ ǫ Z x1 t (ρ f )+ (x 2 , y) 1 + ρ( + (ρ f ) (x 2 , y) − , x 2 )d t ρ + (x 2 ) x1 0 ǫ x1 = (1 + |x 2 |2 + | y|2 )β( , x 2 , y) ǫ Z x1 (ρ f )+ (x 2 , y) + t 1 + ρ( , x 2 )d t ρ (x 2 ) − + ρ (x 2 ) x1 0 ǫ x 1 = (1 + |x 2 |2 + | y|2 )β( , x 2 , y) ǫ x1 + (1 + |x 2 |2 + | y|2 )α1 ( , x 2 , y) ǫ R x1 x (ρ f )+ (x , y) where α1 ( ǫ1 , x 2 , y) := (1+|x |2 +| y|22)ρ+ (x ) ρ + (x 2 ) − x1 0 ρ( ǫt , x 2 )d t . 2 2 1 Using assumptions (B1) and (C1-ii), one can show that α1 is a bounded function which satisfies x (2.2). Since D x 1 V ǫ (x 1 , x 2 , y) = x 1 Fǫ ( ǫ1 , x 2 , y), we derive the result for D x 1 V ǫ (x 1 , x 2 , y). Further, by integrating it, we get ǫ
V (x 1 , x 2 , y) =
x 12 (1 + |x 2 |2
2
+ | y| ) (
ǫ x1
Z )
x1 ǫ
2
tβ1 (t, x 2 , y)d t ,
0
where β1 = α1 + β.
R x1 x Clearly, β2 ( ǫ1 , x 2 , y) := ( xǫ )2 0 ǫ tβ1 (t, x 2 , y)d t is bounded function which satisfies (2.2). The 1 result for the other quantities can be deduced by similar arguments from assumptions (B1), (C1), (C2) and (C3). Lemma 4.8. As ǫ −→ 0,
¯Z ¯¯ ¯ s X r1, ǫ 2, ǫ ǫ ¯ ¯ , X r , Yr ) − f¯(X r1, ǫ , X r2, ǫ , Yrǫ ) d r ¯ → 0 sup ¯ f( ¯ ǫ 0≤s≤t ¯ 0
in probability . ¯R ¯¯ X r1, ǫ ¯ s 2, ǫ ǫ 1, ǫ 2, ǫ ǫ ¯ Proof. We shall show that for every s ∈ [0, t], ¯ 0 f ( ǫ , X r , Yr ) − f (X r , X r , Yr ) d r ¯ tends
to zero in probability as ǫ tends to zero. Let V ǫ denote the solution of equation (4.4). Note that
487
V ǫ has first and second derivatives in (x 1 , x 2 , y) which are possibly discontinuous only at x 1 = 0. Then, as in [15], since ϕ 2 is bounded away from zero, we can use the Itô-Krylov formula to get Zs X r1, ǫ 2, ǫ ǫ ǫ 1, ǫ 2, ǫ ǫ ǫ ǫ , X r , Yr ) − f¯(X r1, ǫ , X r2, ǫ , Yrǫ ) d r V (X s , X s , Ys ) = V (x 1 , x 2 , Y0 ) + f( ǫ 0 Zs + Tr ace a(1) (X r1, ǫ , X r2, ǫ )D2x V ǫ (X r1, ǫ , X r2, ǫ , Yrǫ ) d r Z + Z
2
0 s 0 s
+
[D x 2 V ǫ (X r1, ǫ , X r2, ǫ , Yrǫ )b(1) (X r1, ǫ , X r2, ǫ ) − D y V ǫ (X r1,ǫ , X r2,ǫ , Yrǫ ) f ( [D x V
ǫ
(X r1, ǫ ,
X r2, ǫ , Yrǫ )σ(X r1, ǫ ,
X r2, ǫ ) +
Dy V
ǫ
X r1,ǫ ǫ
(X r1,ǫ , X r2,ǫ , Yrǫ )Z rǫ σ(
0
+ +
1 2 1
Z
s
D2y V ǫ (X r1,ǫ , X r2,ǫ , Yrǫ )Z rǫ σσ∗ ( Z
ǫ
0 s
Dx D y V
2
X r1,ǫ
ǫ
X r1,ǫ
(X r1,ǫ , X r2,ǫ , Yrǫ )σσ∗ (
ǫ
0
X r1,ǫ ǫ
, X r2,ǫ , Yrǫ )]d r , X r2,ǫ )]d B r
, X r2,ǫ )(Z rǫ )∗ d r , X r2,ǫ )(Z rǫ )∗ d r
(4.5)
In view of Lemma 4.7 and Corollary 4.3, V ǫ (x 1 , x 2 , Y0ǫ ) tends to zero as ǫ → 0. Using the fact taht 1 = 1{|X 1, ǫ | 0, let Dn := s : s ∈ [0, t] / |X s | ≤ . ½ ¾n 1 Define also Dn := s : s ∈ [0, t] / |X s1 | ≤ . n Then, there exists a constant c > 0 such that for each n ≥ 1, ǫ > 0, IE|Dnǫ | ≤
c
IE|Dn | ≤
and
n
c n
,
where |. | denotes the Lebesgue measure. Proof. Consider the sequence (Ψn ) of functions defined as follows, x − n − 2n1 2 i f x ≤ − 1n 2 x Ψn (x) = i f − 1n ≤ x ≤ 1n 2 x n
−
1 2n2
x ≥ 1/n
if
¯ := a¯00 := ρ(x 1 , x 2 )−1 . We put, ϕ Using Itô’s formula, we get Z Ψn (X s1 )
=
s ′
¯ s1 , Ψn (X s1 )ϕ(X
Ψn (X 01 ) +
X s2 )dWs
+
0
489
1 2
Z
s 0
¯ 2 (X s1 , X s2 )ds, s ∈ [0, t] Ψ”n (X s1 )ϕ
¯ is lower bounded by C1 , taking the expectation, we get Since ϕ Z t Z t 1[− 1 , 1 ] (X s1 )ds ≤ IE
C1 IE
n n
0
¯ 2 (X s1 , X s2 )ds Ψ”n (X s1 )ϕ 0
= 2IE Ψn (X t1 ) − Ψn (x 1 ) It follows that IE(|Dn |) ≤ 2C1−1 IE Ψn (X t1 ) − Ψn (x 1 ) ≤ c/n. The same argument, applies to Dnǫ , allows us to show the first estimate. Lemma 4.11. Consider a collection {Z ǫ , ǫ > 0} of real valued random variables, and a real valued random variable Z. Assume that for each n ≥ 1, we have the decompositions Z ǫ = Zn1,ǫ + Zn2,ǫ Z = Zn1 + Zn2 , such that for each fixed n ≥ 1, Zn1,ǫ ⇒ Zn1 c IE|Zn2,ǫ | ≤ p n c 2 IE|Zn | ≤ p . n Then Z ǫ ⇒ Z, as ǫ → 0. Proof. The above assumptions imply that the collection of random variables {Z ǫ , ǫ > 0} is tight. Hence the result will follow from the fact that IEΦ(Z ǫ ) → IEΦ(Z),
as ǫ → 0
for all Φ ∈ C b (IR) which is uniformly Lipschitz. Let Φ be such a function, and denote by K its Lipschitz constant. Then |IEΦ(Z ǫ ) − IEΦ(Z)| ≤ IE|Φ(Z ǫ ) − Φ(Zn1,ǫ )| + +|IEΦ(Zn1,ǫ ) − IEΦ(Zn1 )| + IE|Φ(Zn1 ) − Φ(Z)| c ≤ |IEΦ(Zn1,ǫ ) − IEΦ(Zn1 )| + 2K p . n Hence
c lim sup |IEΦ(Z ǫ ) − IEΦ(Z)| ≤ 2K p , n ǫ→0 for all n ≥ 1. The result follows. Proof of Lemma 4.9. For each n ≥ 1, define a function θn ∈ C(IR, [0, 1]) such that θn (x) = 0 for 1 |x| ≤ 2n , and θn (x) = 1 for |x| ≥ 1n . We have Z t Z t Z t 1,ǫ 2,ǫ ǫ 1,ǫ 1,ǫ 2,ǫ ǫ f¯(X 1,ǫ , X 2,ǫ , Y ǫ )[1 − θn (X 1,ǫ )]ds f¯(X , X , Y )θn (X )ds + f¯(X , X , Y )ds = s
s
s
s
0
=
Z
t
f¯(X s1 , X s2 , Ys )ds
0 Zn1,ǫ Z t
=
s
s
s
0 Zn1
s
s
0
+ Zn2,ǫ
Z
t
f¯(X s1 , X s2 , Ys )θn (X s1 )ds +
=
0
s
0
+
Zn2 490
f¯(X s1 , X s2 , Ys )[1 − θn (X s1 )]ds
s
Note that the mapping
Z 1
t
f¯(x s1 , x s2 , ys )θn (x s1 )ds
2
(x , x , y) 7−→
0
is continuous from C([0, t]) × D([0, t]) equipped with the product of the sup–norm and the S topologies into IR. Hence from Proposition 4.5, Zn1,ǫ =⇒ Zn1 as ǫ → 0, for each fixed n ≥ 1. Moreover, from Lemma 4.10, the linear growth property of f¯, Lemma 4.1 and Proposition 4.2, we deduce that c E|Zn2,ǫ | ≤ p , n
c E|Zn2 | ≤ p . n
Lemma 4.9 now follows from Lemma 4.11. Proof of Proposition 4.6 Passing to the limit in the backward component of the equation (1.4) and using Lemmas 4.8 and 4.9, we derive assertion (i). Assertion (ii) can be proved by using the same arguments as those in section 6 of [24].
4.3 Identification of the limit martingale. Since f¯ is uniformly Lipschitz in y and H is bounded, then standard arguments of BSDEs (see e. g. [23]) show that the BSDE (1.5) has a strongly unique solution and we have, Proposition 4.12. Let (Y¯s , Z¯s , 0 ≤ s ≤ t) be the unique solution to BSDE (1.5). Then, for every s ∈ [0, t], Z. Z. IE|Ys − Y¯s |2 + IE [M − Z¯r d M X ] t − [M − Z¯r d M X ]s = 0. r
r
0
0
Proof. For every s ∈ [0, t] \ D, we have Rt ¯ Ys = H(X t ) + s f (X r , Yr )d r − (M t − Ms )
Y¯s = H(X t ) +
Rt s
f¯(X r , Y¯r )d r −
Rt s
Z¯r d M rX
R. ¯ := Z¯r d M X is a Fs -martingale. Arguing as in [24], we show that M r s Since f¯ satisfies condition (C1), we get by Itô’s formula, that Z. Z. Z t 2 X X IE|Ys − Y¯s | + IE [M − Z¯r d M r ] t − [M − Z¯r d M r ]s ≤ CIE |Yr − Y¯r |2 d r. 0
0
s
Therefore, Gronwall’s lemma yields that IE|Ys − Y¯s |2 = 0, ∀s ∈ [0, t] − D. ¯ Since Y¯ is continuous, Y is càd-lag and ∀s ∈ [0, t]. Z D is countable, then Z Ys = Ys , IP-a.s, .
Moreover, we deduce that, IE [M −
0
.
Z¯r d M rX ] t − [M −
491
Z¯r d M rX ]s 0
= 0.
As a consequence of Proposition 4.12, we have Z· Z. l aw ǫ ǫ Xǫ X Corollary 4.13. Y , =⇒ Y¯ , Zr d Mr Z¯r d M r . 0
0
Theorem 3.3 is proved.
5 Proof of Theorem 3.4. Since the SDE (3.1) is weakly unique ([18]), the martingale problem associated to X = (X 1 , X 2 ) is well posed. We then have the following: Proposition 5.1. (i) For any t > 0, x ∈ IRd , the BSDE Z t Z t x x t, x t, x x f¯(X r , Yr )d r − Z rt, x d M rX , 0 ≤ s ≤ t. Ys = H(X t ) + s
s
t, x
admits a unique solution (Yst, x , Zst, x )0≤s≤t such that the component (Yst, x )0≤s≤t is bounded and Y0 is deterministic. t, x
(ii) If moreover, the deterministic function, (t, x) ∈ [0, T ] × IRd+1 7−→ v(t, x) := Y0 C [0, T ] × IRd+1 , IR , then it is a L p -viscosity solution of the PDE (3.2).
belongs to
t, x
Remark. The continuity of the map (t, x) 7−→ v(t, x) := Y0 , which is assumed in assertion (ii) of Propostion 5.1, will be established in Proposition 5.3 below. Proof of Proposition 5.1. (i) Thanks to Remark 3.5 of [23], it is enough to prove existence and uniqueness for the BSDE Z t Z t t, x x x t, x Y = H(X ) + f¯(X , Y )d r − Z t, x d B r , 0 ≤ s ≤ t. s
t
r
r
s
r
s
Since f satisfies (C) and ρ is bounded, one can easily verify that f¯ is uniformly Lipschitz in y uniformly with respect to (x 1 , x 2 ) and satisfies (C1)-(ii). Existence and uniqueness of solution follow then from standard results for BSDEs, see e. g. [22]. Moreover, since H is uniformly bounded and f¯ satisfies the linear growth condition (C1)-(ii), one can prove that the solution Y t, x is bounded, t, x see e. g. [1]. Finally, since (Yst, x ) is FsX −adapted then Y0 is measurable with respect to a trivial σ−algebra and hence it is deterministic. t,x (ii) Assume that the function v(t, x) := Y0 belongs to C [0, T ] × IRd+1 , IR . We only prove that v is a L p –viscosity sub–solution. The proof of the super–solution property can be done similarly. Since the coefficient of PDE under consideration are time homogeneous, then v(t, x) is solution to the initial value problem (1.6) if and only if the function u(t, x) := v(T − t, x) is solution to the terminal value problem. ∂ u (t, x) = ( ¯L u)(t, x) + f¯(x, u(t, x)) ∂t u(T, x) = H(x). 492
t ∈ [0, T ],
(5.1)
Working with this backward PDE will simplify the details of the proofs below. Let X st,x be the unique weak solution to SDE (3.1). We will establish that the solution Y of the Markovian BSDE Z T Z T t,x t,x t, x t,x t, x Ys = H(X T ) + 0 ≤ t ≤ s ≤ T. (5.2) f¯(X r , Yr )d r − Z rt, x d M rX , s
s
t,x
define a L p −viscosity sub–solution to the problem (5.1) by puting u(t, x) := Yt . 1, 2 Let ϕ ∈ Wp, l oc [0, T ] × IRd+1 , IR , let (bt , b x ) ∈ [0, T ] × IRd+1 be a point which is a local maximum of u − ϕ. Since p > d + 2, then ϕ has a continuous version which we consider from now on. We assume without loss of generality that v(bt , b x ) = ϕ(bt , b x)
(5.3)
We will argue by contradiction. Assume that there exists ǫ, α > 0 such that ∂ϕ ∂s
(s, x) + ¯L ϕ(s, x) + f¯(x, u(s, x)) < −ǫ, λ–a.e. in Bα (bt , b x ).
(5.4)
where λ denote the Lebesgue measure. Since (bt , b x ) is a local maximum of u − ϕ, we can find a positive number α′ (which we can suppose equal to α) such that x) (5.5) u(t, x) ≤ ϕ(t, x) in Bα (bt , b Define
n τ = inf s ≥ bt , ;
o |X sbt , bx − b x | > α ∧ (bt + α)
Since X is a Markov diffusion and f¯ is uniformly Lipschitz in y and satisfies condition (C1)-(ii), then arguing as in [10], one can show that for every r ∈ [bt , bt + α], Yrbt , bx = u(r, X rbt , bx ). Hence, the bt , b x process (Y¯s , Z¯s ) := ((Ys∧τ ), 11[0, τ] (s)(Z bt , bx ))s∈[bt , bt +α] solves the BSDE s
Z Y¯s
=
bt +α
X τbt , bx ) +
u(τ,
11[0, τ] f¯(r, X rbt , bx , u(r, X rbt , bx ))d r s
Z
bt +α
Z¯r d M rX
−
bt , b x
s ∈ [bt , bt + α].
,
s
bs )s∈[bt , bt +α] , defined by (Ybs , Z bs ) := On other hand, by Itô-Krylov formula, the process (Ybs , Z bt , b x bt , b x ϕ(s ∧ τ, X s∧τ ), 11[0, τ] (s)∇ϕ(s, X s ) solves the BSDE Z Ybs
=
ϕ(τ, Z −
X τbt , bx ) −
bt +α
br d M X Z r
bt +α
11[0, τ] [( s bt , b x
∂ϕ ∂r
+ ¯L ϕ)(r, X rbt , bx )]d r
.
s
From the choice of τ, (τ, X τbt , bx ) ∈ Bα (bt , b x ). Therefore, u(τ, X τbt , bx ) ≤ ϕ(τ, X τbt , bx ). 493
∂ϕ Let A := {(t, x) ∈ Bα (bt , b x ), [ ∂ s + ¯L ϕ + f¯(., u(.))](t, x) < −ǫ} and A¯ := Bα (bt , b x ) \ A the complement ¯ of A. By (5.4), λ(A) = 0.
Since the diffusion {X sˆt ,ˆx , s ≥ t} is nondegenerate, Krylov’s inequality ([17], Ch. 2, Sec. 2 & 3) implies that 11A¯(r, X rbt , bx ) = 0 d r × dIP− a.e. It follows that Z
bt +α
IE bt
−11[0, τ] [(
∂ϕ ∂r
+ ¯L ϕ)(r, X rbt , bx ) + f¯(r, X rbt , bx , u(r, X rbt , bx ))])d r ≥ IE(τ − bt ) ǫ > 0
(5.6)
∂ϕ This implies that [−11[0, τ] [( ∂ r + ¯L ϕ)(r, X rbt ,bx ) + f¯(r, X rbt ,bx , u(r, X rbt ,bx ))])] > 0 on a set of d t × dIP positive measure. Therefore, the strict comparison theorem (Remark 2.5 in [23]) shows that Y¯bt < Ybbt , that is u(bt , b x ) < ϕ(bt , b x ), which contradicts our assumption (5.3).
Under assumptions (A), (B), the SDE (3.1) has a unique weak solution, see [18]. We then have the following continuity property. Proposition 5.2. (Continuity in law of the map x 7→ X .x ) Assume (A), (B). Let X sx be the unique weak solution of the SDE (3.1), and Zs Zs n n ¯b(X )d r + ¯ n )d B r , 0 ≤ s ≤ t X := x n + σ(X s
r
r
0
0
l aw
Assume that x n → x = (x 1 , x 2 ) ∈ IR1+d as n → ∞. Then X n =⇒ X x . ¯ satisfy (A), (B), one can easily check that the sequence X n is tight in C ([0, t]× Proof. Since ¯b and σ IRd+1 ). By Prokhorov’s theorem, there exists a subsequence (denoted also by X n ) which converges weakly to a process Xb . We shall show that Xb is a weak solution of SDE (3.1). • Step 1: For every ϕ ∈ Cc∞ (IR1+d ), Zu ¯L ϕ(Xbv )d v is a F Xb -martingale. b ∀u ∈ [0, t], ϕ(X r ) − 0
All we need to show is that for every ϕ ∈ Cc∞ (IR1+d ), every 0 ≤ s ≤ u and every function Φs of x (X r n )0≤r d + 2, the function v(t, x) := Y0
is a L p -viscosity solution to the PDE (1.6).
Proof. (i) Let Y t,x be the limit process defined in Proposition 4.5. We have Z t X 1,ǫ ǫ ǫ Y0 = H(X t ) + f ( r , X r2, ǫ , Yrǫ )d r − M tǫ ǫ 0
Z t,x Y0
=
t
H(X tx ) + 0
f¯(X rx , Yrt,x )d r − M t
From Jakubowski [14], the projection: y 7→ y t is continuous from D([0, t]; IR) into IR for the S– topology. We then deduce from the convergence of the above right–hand sides that Y0ǫ converges towards Y0 in distribution. Since Y0ǫ and Y0 are deterministic, this means exactly that Y0ǫ → Y0 (ii) Let (t n , x n ) → (t, x). We assume that t > t n > 0. We have, Z tn Z tn xn xn tn, xn x t , x Z rt n , x n d M rX , 0 ≤ s ≤ t n , Ys = H(X t n ) + f¯(X r n , Yr n n )d r − s
(5.7)
s
l aw
where X x n ⇒ X x . Since H isRa bounded continuous function and f¯ satisfies (C1), one can easily show that the sequence . xn x {(Y t n , x n , 0 1[s,t n ] (u)Z r n d M rX )}n∈IN∗ is tight in D([0, t]; IR2 ). Let us rewrite the equation (5.7) as follows Z t Z t xn xn tn, xn t , x x = H(X ) + Y (5.8) 1[s,t ] (u)Z t n , x n d M X f¯(X n , Y n n )d r − tn
s
Z − =
r
r
s
n
s
t
f¯(X rx n , Yrt n , x n )d r, 0 ≤ s ≤ t.
tn 1 An + A2n
495
r
r
• Convergence of A2n
¯ ¯Z ¯ ¯ t ¯ ¯ f¯(X rx n , Yrt n , x n )d r ¯ ≤ K|t − t n |. Hence A2n tends to zero in probability. Since f¯ is bounded, IE ¯ ¯ ¯ t n
• Convergence of A1n R. xn x Denote by (Y ′ , M ′ ) the weak limit of {(Y t n , x n , 0 1[s,t n ] (u)Z r n d M rX )}n∈IN∗ . The same proof as that Z t Z t l aw xn tn, xn ¯ of Lemma 4.9 establishes that f (X , Y )d r =⇒ f¯(X x , Y ′ )d r. r
r
r
s
r
s
Passing to the limit in (5.8), we obtain that Z Ys′
=
t
H(X tx ) + s
f¯(X rx , Yr′ )d r − (M t′ − Ms′ ), s ∈ [0, t] ∩ D c . l aw
The uniqueness of the considered BSDE ensures that ∀s ∈ [0, t], Ys′ = Yst, x IP-ps. Hence Y t n , x n ⇒ t , x n l aw
Y t, x . As in (i), one derive that Y0 n
t, x
=⇒ Y0
t, x
which yields to the continuity of Y0 .
Assertion (iii) follows from (ii) and the second statement of Proposition 5.1.
A Appendix: S-topology The S–topology has been introduced by Jakubowski ([14], 1997) as a topology defined on the Skorohod space of càdlàg functions: D([0, T ]; IR). This topology is weaker than the Skorohod topology but tightness criteria are easier to establish. These criteria are the same as the one used in Meyer-Zheng [20]. Let N a, b (z) denotes the number of up-crossing of the function z ∈ D([0, T ]; IR) from level a to level b (a < b). We recall some facts about the S–topology. Proposition A.1. (A criteria for S-tight). A sequence (Y ǫ )ǫ>0 is said to be S–tight if and only if it is relatively compact for the S–topology. Let (Y ǫ )ǫ>0 be a family of stochastic processes in D([0, T ]; IR). Then this family is tight for the S– topology if and only if (kY ǫ k∞ )ǫ>0 and (N a, b (Y ǫ ))ǫ>0 are tight for each a < b. Let Ω, F , IP, (F t ) t≥0 be a stochastic basis. If (Y )0≤t≤T is a process in D([0, T ]; IR) such that Yt is integrable for any t, the conditional variation of Y is defined by C V (Y ) =
sup
n−1 X
n≥1, 0≤t 1 0
sup |Ytǫ |
sup |Yt |
.
t∈[0, T ]
< +∞.
t∈[0, T ]
Theorem A.2. Let (Y ǫ )ǫ>0 be a S-tight family of stochastic process whose trajectories belong to D([0, T ]; IR). Then there exists a sequence (ǫk )k∈IN decreasing to zero, some process Y ∈ D([0, T ]; IR) and a countable subset D ∈ [0, T ] such that for any n ≥ 1 and any (t 1 , ..., t n ) ∈ [0, T ]\D, ǫ
ǫ
Dist
(Yt 1k , ..., Yt nk ) −→ (Yt 1 , ..., Yt n ) Remark A.3. The projection π T : y ∈ (D([0, T ]; IR), S) 7→ y(T ) is continuous (see Remark 2.4, p.8 in Jakubowski [14]), but y 7→ y(t) is not continuous for each 0 ≤ t ≤ T . Lemma A.4. Let (U ǫ , M ǫ ) be a multidimensional process in D([0, T ]; IR p ) (p ∈ IN∗ ) converging to ǫ (U, M ) in the S-topology. Let (F tU ) t≥0 (resp. (F tU ) t≥0 ) be the minimal completeadmissible filtration generated by U ǫ (resp. U). We assume moreover that for every T > 0, supǫ>0 IE sup0≤t≤T |M tǫ |2 < CT . ǫ If M ǫ is a F U -martingale and M is F U -adapted, then M is a F U -martingale. ǫ Lemma A.5. Let of process converging weakly in D([0, T]; IR p ) to Y . We assume (Y )ǫ>0 be a sequence that supǫ>0 IE sup0≤t≤T |Ytǫ |2 < +∞. Then for any t ≥ 0, E sup0≤t≤T |Yt |2 < +∞.
Acknowlegement. The authors are sincerely grateful to the referee for many useful remarks which have leaded to an improvement of the paper. The second author thanks IMATH laboratorie of université du Sud Toulon-Var and the LATP laboratory of université de Provence, Marseille, France, for their kind hospitality.
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