B hold. Then, from Khasminskii Krylov [15] and Krylov [18], we deduce

3 The main results Consider the equation X x t = x + Z t ¯bX x s ds + Z t ¯ σX x s d B s , t ≥ 0. 3.1 Assume that A, B hold. Then, from Khasminskii Krylov [15] and Krylov [18], we deduce that for each fixed, x ∈ IR d+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 intro- duced by Caffarelli et al. in [7] to study PDEs with measurable coefficients. Presentations of this topic can be found in [7; 8]. Let g : IR d+1 × IR 7−→ IR be a measurable function and ¯L := X i, j ¯ a i j x ∂ 2 ∂ x i ∂ x j + X i ¯b i x ∂ ∂ x i denote the second order PDE operator associated to the SDE 3.1. We consider the parabolic equation    ∂ v ∂ t t, x = ¯L vt, x + gx, vt, x, t ≥ 0 v0, x = Hx. 3.2 Definition 3.1. Let p be an integer such that p d + 2. a A function v ∈ C € [0, T ] × IR d+1 , IR Š is a L p -viscosity sub-solution of the PDE 3.2, if for every x ∈ IR d+1 , v0, x ≤ Hx and for every ϕ ∈ W 1, 2 p, l oc € IR + × IR d+1 , IR Š and bt, bx ∈ 0, T] × IR d+1 at which v − ϕ has a local maximum, one has ess lim inf t, x→bt, bx ½ ∂ ϕ ∂ t t, x − ¯Lϕt, x − gx, vt, x ¾ ≤ 0. b A function v ∈ C € [0, T ] × IR d+1 , IR Š is a L p -viscosity super-solution of the PDE 3.2, if for every x ∈ IR d+1 , v0, x ≥ Hx and for every ϕ ∈ W 1, 2 p, l oc € IR + × IR d+1 , IR Š and bt, bx ∈ 0, T] × IR d+1 at which v − ϕ has a local minimum, one has ess lim sup t, x→bt, bx ½ ∂ ϕ ∂ t t, x − ¯Lϕt, x − gx, vt, x ¾ ≥ 0. Here, Gt, x, ϕs, x is merely assumed to be measurable upon the variable x =: x 1 , x 2 . c A function v ∈ C € [0, T ] × IR d+1 , IR Š is a L p -viscosity solution if it is both a L p -viscosity sub- solution and super-solution. 483 Remark 3.2. Condition a means that for every ǫ 0, r 0, there exists a set A ⊂ B r bt, bx of positive measure such that, for every s, x ∈ A, ∂ ϕ ∂ s s, x − ¯Lϕt, x − gx, vt, 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 + × IR d+1 , there exists a process X s , Y s , Z s ≤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]; IR d+1 equipped with the uniform topology. ii the sequence of processes Y ǫ s , R t s Z ǫ r d M X ǫ r ≤s≤t converges in law to the process Y s , R t s Z r d M X r ≤s≤t in D[0, t]; IR 2 , 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, a Y,Z is F X −adapted and Y s , R t s Z r d M X r ≤s≤t is continuous. b IE sup ≤s≤t |Y s | 2 + R t |Z r σX 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 a-b then, IE sup ≤s≤t ¯ ¯Y 1 s − Y 2 s ¯ ¯ 2 + R t ¯ ¯Z 1 r σX r − Z 2 r σX r ¯ ¯ 2 d r = 0, i. e. since σσ ∗ is elliptic see A3, Y 1 s = Y 2 s ∀0 ≤ s ≤ t, IP a. s., and Z 1 s = Z 2 s 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 Y t,x s s be the unique solution of the BSDE 1.5. Then i Equation 1.6 has a unique L p -viscosity solution v such that vt, x = Y t,x . ii For every t, x ∈ IR + × IR d+1 , v ǫ t, x → vt, x, as ǫ → 0. 4 Proof of Theorem 3.3. In all of this section, t, x ∈ IR + × IR d+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 ≤s≤t [|X 1, ǫ s | p + |X 2, ǫ s | p ] Œ ≤ C p . 484

4.1 Tightness and convergence for the BSDE.