Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs

In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by solutions of penalized partial differential equations.


Introduction
Let G be a C 2 convex, open and bounded set from R d and for (t, x) ∈ [0, T ] ×Ḡ we consider the following reflecting stochastic differential equation (SDE for short) with K a bounded variation process such that for any s ∈ [t, T ], K s = s t ∇ℓ(X r )d |K| [t,r] and |K| [t,s] = s t 1 {Xr∈∂G} d |K| [t,r] , where the notation |K| [t,s] stands for the total variation of K on the interval [t, s].
The coefficients b and σ are supposed to be only bounded continuous on R d and σσ * uniformly elliptic. The first main purpose is to prove that the weak solution (X, K) is approximated in law (in the space of continuous functions) by the solutions of the non-reflecting SDE X n s = x + s t [b(X n r )−n(X n r − πḠ(X n r ))] dr + s t σ(X n r )dW r , s ∈ [t, T ] , where πḠ is the projection operator. Since for n → ∞ the term K n s := n s t (X n r − πḠ(X n r ))dr forces the solution X n to remain near the domain, the above equation is called SDE with penalization term.
The case where b and σ are Lipschitz has been considered by Lions, Menaldi and Sznitman in [10] and by Menaldi in [13] where they have proven that E(sup s∈[0, T ] |X n s − X s |) −→ 0, as n → ∞. Note that Lions and Sznitman have shown, using Skorohod problem, the existence of a weak solution for the SDE with normal reflection to a (non-necessarily convex) domain. The case of reflecting SDE with jumps has been treated by Łaukajtys and Słomiński in [7] in the Lipschitz case; the same authors have extended in [8] these results to the case where the coefficient of the reflecting equation is only continuous. In these two papers it is proven that the approximating sequence (X n ) n is tight with respect to the S-topology, introduced by Jakubowski in [5] on the space D R + , R d of càdlàg R d -valued functions. Assuming the weak (in law) uniqueness of the limiting reflected diffusion X, they prove in [8] that X n S-converges weakly to X. We mention that (X n ) n may not be relatively compact with respect to the Skorohod topology J 1 .
In contrast to [8], we can not simply assume the uniqueness in law of the limit X, and the weak S-convergence of X n to X is not sufficient to our goal. In our framework, we need to show the uniqueness in law of the couple (X, K) and that the convergence in law of the sequence (X n , K n ) to (X, K) holds with respect to uniform topology.
The first main result of our paper will be the weak uniqueness of the solution (X, K), together with the convergence in law (in the space of continuous functions) of the penalized diffusion to the reflected diffusion X and the continuity with respect to the initial data.
Subsequently, using a proper generalized BSDE, we deduce (as a second main result) an approximation result for a continuous viscosity solution of the system of semi-linear partial differential equations (PDEs for short) with a nonlinear Neumann boundary condition where L is the infinitesimal generator of the diffusion X, defined by and ∂u i /∂n is the outward normal derivative of u i on the boundary of the domain. Boufoussi and Van Casteren have established in [2] a similar result, but in the case where the coefficients b and σ are uniformly Lipschitz.
We mention that the class of BSDEs involving a Stieltjes integral with respect to the continuous increasing process |K| [t,s] was studied first in [15] by Pardoux and Zhang; the authors provided a probabilistic representation for the viscosity solution of a Neumann boundary partial differential equation. It should be mentioned that the continuity of the viscosity solution is rather hard to prove in our frame. In fact, this property essentially use the continuity with respect to initial data of the solution of our BSDE. We develop here a more natural method based on the uniqueness in law of the solution (X, K, Y ) of the reflected SDE-BSDE and on the continuity property. Similar techniques were developed, in the non reflected case, in [1], but in our situation the proof is more delicate. The difficulty is due to the presence of the reflection process K in the forward component and the generalized part in the backward component.
Throughout this paper we use different types of convergence defined as follows: for the processes (Y n ) n and Y , by Y n * − −− → u Y we denote the convergence in law with respect to the uniform topology, by Y n * − −− → J 1 Y n we mean the convergence in law with respect to the Skorohod topology J 1 and by Y n * − −− → S Y we understand the weak convergence considered in S-topology. The paper is organized as follows: in the next section we give the assumptions, we formulate the problem and we state the two main results. The third section is devoted to the proof of the first main result (proof of the convergence in law of (X n , K n ) to (X, K) as n → ∞ and the continuous dependence with respect to the initial data). In Section 4 the generalized BSDEs are introduced, the continuity with respect to the initial data is obtained and we prove the approximation result for the PDE introduced above.

Formulation of the problem; the main results
Let G be a C 2 convex, open and bounded set from R d and we suppose that there exists a function and, for all x ∈ ∂G, ∇ℓ (x) is the unit outward normal to ∂G. In order to define the approximation procedure we shall introduce the penalization term. Let p : R d → R + be given by p (x) = dist 2 (x,Ḡ).
Without restriction of generality we can choose ℓ such that where δ (x) := ∇p (x) is called the penalization term. It can be shown that p is of class C 1 on R d with where πḠ (x) is the projection of x onḠ. It is clear that δ is a Lipschitz function. On the other hand, x → dist 2 x,Ḡ is a convex function and therefore Let T > 0 and suppose that:

Remark 1
In fact we can assume that the functions b and σ have sublinear growth but, for the simplicity of the calculus, we will work with assumption (A 1 ).
Moreover, there exist some positive constants C i , i = 1, 2, α ∈ R, β ∈ R * + and q ≥ 1 such that Let us consider the following system of semi-linear PDEs considered on the whole space: and the next semi-linear PDE considered with Neumann boundary conditions: where L is the second order partial differential operator and, for any x ∈ ∂G is the exterior normal derivative in x ∈ ∂G.
Our goal is to establish a connection between the viscosity solutions for (3) and (4) respectively. The proof will be given using a probabilistic approach. Therefore we start by studying an SDE with reflecting boundary condition and then we associate a corresponding backward SDE. Since the coefficients of the forward equation are merely continuous, our setting is that of weak formulation of solutions.
For (t, x) ∈ [0, T ] ×Ḡ we consider the following stochastic differential equation with reflecting boundary condition: where K t,x [t,s] is the the total variation of K t,x on the interval [s, t] * . We denote by k t,x s the continuous increasing process defined by k t,x Using the penalization term δ we can define the approximation procedure for the reflected diffusion X.
Here and subsequently, we shall denote by V and V n : Hence (5) and (7) become respectively

Definition 2
We say that Ω, F, P, {F s } s≥t , W, X, K is a weak solution of (5) if Ω, F, P, {F s } s≥t is a stochastic basis, W is a d ′ -dimensional Brownian motion with respect to this basis, X is a continuous adapted process and K is a continuous bounded variation process such that X s ∈Ḡ, ∀s ∈ [t, T ] and system (5) is satisfied.
The main results are the following two theorems. The first one consists in establishing the weak uniqueness (in law) of the solution for (5) and the continuous dependence in law with respect to the initial data.

Theorem 3 Under the assumptions
is continuous in law.
Once this result for the forward part is established we then associate a BSDE involving Stieltjes integral with respect to the increasing process k t,x in order to obtain the probabilistic representation for the viscosity solution of PDE (3).
The next result provides the approximation of a viscosity solution for system (4) by the solutions sequence of (3).

Proof of Theorem 3
We shall divide the proof of this Theorem into several lemmas. First of all we recall that the existence of a weak solution is given, under assumption (A 1 ), by [11,Theorem 3.2].
For the simplicity of presentation we suppress from now on the explicit dependence on (t, x) in the notation of the solution of (5) and (7).
We first prove an estimation result for the solutions of the penalized SDE (7).

Lemma 5
Under assumption (A 1 ), for any q ≥ 1, there exists a constant C > 0, depending only on d, T and q, such that Proof. Without loss of generality we can assume that 0 ∈ G. From Itô's formula applied for |X n s | 2 it can be deduced that Write τ m := inf {s ∈ [t, T ] : |X n s | ≥ m} ∧ T , m ∈ N * and by the above, Here and in what follows C > 0 will denote a generic constant which is allowed to vary from line to line. Therefore By Burkholder-Davis-Gundy inequality we deduce and (11) yields since from (1) applied for z = 0 ∈ G, we have s t X n r , dK n r = n s t X n r , δ(X n r ) dr ≥ 0.
From the Gronwall lemma, Taking m → ∞ it follows that Once again from (11) and (12) we obtain We have that there exists ε > 0 such that the ballB (0, ε) ⊂ G, and, for z = ε and by the definition of total variation of K n , it follows that

Lemma 6
Under assumption (A 1 ) the sequence (X n s , K n s , k n s ) s∈[t,T ] is tight with respect to the Stopology.
Proof. In order to obtain the S-tightness of a sequence of integrable càdlàg processes U n , n ≥ 1, we shall use the sufficient condition given e.g. in [9, Appendix A] which consists in proving the uniform boundedness for: defines the conditional variation of U n , with the supremum taken over all finite partitions π : t = t 0 < t 1 < · · · < t m = T. Using Lemma 5, we deduce that there exists a constant C > 0 such that for every n ∈ N * Since k n is increasing and l ∈ C 2 b R d , then there exist constants M, C > 0 such that for every n ∈ N * By the definition of V n , assumption (A 1 ) and the fact the conditional variation of a martingale is 0, we obtain for each n ∈ N * , Therefore (see also Lemma 5), there exists C > 0 such that for every n ∈ N * |X n s | ≤ C.

Lemma 7
Under the assumptions (A 1 −A 2 ), the uniqueness in law of the stochastic process (X s ) s∈[t,T ] holds.

Lemma 9
We suppose that the assumptions (A 1 − A 2 ) are satisfied. Then

Proof.
(i) First we will prove the convergence: We shall apply [8, Theorem 4.3 (iii)]. We recall that we have the uniqueness of the weak solution. For any n ∈ N, s ∈ [t, T ], let H n s := x ∈Ḡ and the processes Z n s := (s, W s ). Our equation can be written as The processes Z n satisfies the (UT) condition (introduced in [18]), since for any discrete predictable processes U n ,Ū n of the form U n s := U n 0 + k i=0 U n i , respectivelyŪ n s :=Ū n 0 + Therefore the assumptions of [8,Theorem 4.3] are satisfied and thus we obtain that Using once again [8, Theorem 4.3 (ii)] and definition (9) we deduce that for any partition t = t 0 < t 1 < · · · < t m = T. The above convergence is considered in law, on the space R d m × D([0, T ] , R d ) endowed with the product between the usual topology on R d m and the Skorohod topology J 1 . Hence since (X n , V n ) n is tight. It is known that the space D([0, T ] , R d ) of càdlàg functions endowed with S-topology is not a linear topological space, but the sequential continuity of the addition, with respect to the S-topology, is fulfilled (see Jakubowski [5,Remark 3.12]). Therefore In order to obtain the uniform convergence § of the sequence (X n , K n ) n we remark that, V and V n , V are continuous, this convergence is uniform in distribution: Using the Skorohod theorem, there exists a new probability space Ω ,F ,P on which we can define random variablesV ,V n such that LetX n be the solution of the equation It is easy to prove (see, e.g., [7,Lemma 2.2] or [20,Lemma 2.2]) that Since (X n ,K n ) law = == = (X n , K n ) and |K n | [t,T ] is bounded in probability by inequality (10), |K n | [t,T ] is bounded in probability. Applying [7,Theorem 2.7], it follows that |K n | [t,T ] is also bounded in probability.
But sup therefore, from (16), sup On the other hand, letX be the solution of the Skorohod problem Returning to the proof of Lemma 9, the conclusion (ii) follows now easily, since k and k n are defined by (6) and (8) respectively.

Remark 11
Let the assumptions (A 1 −A 2 ) be satisfied. Then the weak solution (X t,x s ) s∈[t,T ] is a strong Markov process. Indeed, taking into account the equivalence between the existence for the (sub-)martingale problem and the existence of a weak solution for reflected SDE (5) (see [4,Theorem 7]), we obtain that the weak solution (X t,x s ) s∈[t,T ] is a strong Markov process since the uniqueness holds (see [4,Theorem 10]). In our situation, this equivalence can be obtained by using Krylov's inequality for reflecting diffusions.
The following result will finalize the proof of Theorem 3. We extend the solution process to [0, T ] by denoting X t,x s := x, K t,x s := 0, ∀s ∈ [0, t).

Lemma 12
We suppose that the assumptions (A 1 −A 2 ) are satisfied and let (X t,x s , K t,x s ) s∈[t,T ] be the weak solution of (5). Then (i) the family (X t,x s , K t,x s ) s∈[0,T ] is tight with respect to the initial data (t, x), as family of C([0, T ],R d × R d )-valued random variables and (ii) the weak solution (X t,x s , K t,x s ) s∈[t,T ] is continuous in law with respect to the initial data (t, x).
Proof. (i) First let (t, x) ∈ [0, T ] ×Ḡ be fixed. Denote as before (X s , K s ) := (X t,x s , K t,x s ). Applying Itô's formula for the process X s − X r , where r is fixed and s ≥ r, we deduce Therefore, using that b, σ are bounded functions andḠ is a bounded domain, Concerning K, we remark first that Observe that the constants in the right hand of the inequalities (19) and (20) do not depend on (t, x).
(ii) Taking into account the conclusion (i) and the Prokhorov theorem, we have that if (t n , x n ) → (t, x), as n → ∞, then there exists a subsequence, still denoted by (t n , x n ), such that It remains to identify the limits, i.e. X law = == = X t,x and K If V n is defined by σ(X n r )dW n r then X n s + K n s = V n s , P-a.s.; using [16, Corollary 2.14], we see that andX n s +K n s =V n s , a.s. which yields, passing to the limit, that X s +K s =V s , a.s.
Then the coupled process (X s ,K s ) is a solution of (5) corresponding to the initial data (t, x). Taking into account the uniqueness in law of the solution (X t,x s , K t,x s ) s∈[t,T ] (see Remark 8) we deduce that the whole sequence (X n s , K n s ) s∈[t,T ] converges to the process (X t,x s , K t,x s ) s∈[t,T ] , and therefore the continuity with respect to (t, x) follows.

BSDEs and nonlinear Neumann boundary problem
Let us now consider the processes (X t,x,n s , k t,x,n s ) t≤s≤T and (X t,x s , k t,x s ) t≤s≤T given by rela- In order to give the proof of Theorem 4 we associate the following generalized backward stochastic differential equations (BSDE for short) on [t, T ]: and respectively the BSDE corresponding to the solution of (7) where are the martingale part of the reflected diffusion process X t,x and X t,x,n respectively. We assume for simplicity that the processes (X t,x,n s , K t,x,n s ) s∈[t,T ] and (X t,x s , K t,x s ) s∈[t,T ] are considered on the canonical space.
We recall that the coefficients f, g and h satisfy assumption (A 3 ). Then, given the processes (X t,x,n s , k t,x,n s ) s∈[t,T ] and (X t,x s , k t,x s ) s∈[t,T ] , this assumption ensures (see [15]) the existence and the uniqueness for the couples (Y t,x,n s , U t,x,n s ) s∈[t,T ] and (Y t,x s , U t,x s ) s∈[t,T ] respectively. Arguing as in [2], one can establish the following result.

Proposition 13 Let the assumptions
T ] be the solutions of the BSDEs (22) and (21), respectively. Then For the proof we will adapt the steps from the proof of [2, Theorem 3.1].
Step 1. The solutions satisfy the boundedness conditions (for the proof see, e.g., [15, where C > 0 is a constant not depending on n. Step 2. To obtain the tightness property with respect to the S-topology it is sufficient to compute the conditional variation CV T (see definition (13)) of the processes Y tn,xn , M tn,xn and H tn,xn respectively; we recall the notation (24) for the quantities M tn,xn and H tn,xn . As in [2, Theorem 3.1], after some easy computation we deduce that there exists a constant C > 0 independent of n, such that In order to pass to the limit in (25)  Since the processesȲ ,M andH are càdlàg, the above equality holds true for any s ∈ [0, T ]. We mention that M X t,x andM are martingales with respect to the same filtration. Indeed,M s is F X t,x ,Ȳ ,M s -adapted and, moreover,M is an F X t,x ,Ȳ ,M -martingale (for the proof see and therefore M X t,x is a F X t,x ,Ȳ ,M -martingale. Now since Y t,x and U t,x are F X t,x -adapted, M t,x := · t U t,x r dM X t,x r is also F X t,x ,Ȳ ,Mmartingale. and, as in the the proof of Proposition 15, we deduce that the limit of u(t n , x n ) is It is easy to show that, even if b and σ are only continuous functions, the proof from [15,Theorem 4.3] (see also [14,Theorem 3.2] for nonreflecting case) still works in order to show that the functions u n and u defined by (27) are viscosity solutions of the PDEs (3) and (4) respectively. Finally, as a consequence of Proposition 13 we deduce the solution u of the deterministic system (4) is approximated by the functions u n , i.e.