Stability of solutions of BSDEs with random terminal time

March 2006, Vol. 10, p. 141–163 www.edpsciences.org/ps

DOI: 10.1051/ps:2006006

STABILITY OF SOLUTIONS OF BSDES WITH RANDOM TERMINAL TIME

Sandrine Toldo

Abstract. In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if (Wⁿ) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if (τⁿ) is a sequence of stopping times that converges to a stopping timeτ, then the solution of the BSDE driven byWⁿwith random terminal timeτⁿconverges to the solution of the BSDE driven byW with random terminal timeτ.

Mathematics Subject Classification. 60H10, 60Fxx, 60G40.

Received November 25, 2004. Revised April 10 and June 23, 2005.

Introduction

We want to make an approximation of the solutions of a backward stochastic differential equation (BSDE for short) with an almost surely finite random terminal timeτ like

Y_t∧τ =ξ+ _τ

t∧τ

f(s, Y_s, Z_s)ds− _τ

t∧τ

Z_sdW_s, t0.

The robustness of numerical methods to approximate solutions of BSDEs had already been studied in the case of a deterministic terminal time. Antonelli and Kohatsu-Higa in [1] and also Coquet, Mackeviˇcius and M´emin in [6] and [7] proposed approximation schemes that use discretization of filtrations and convergence of filtrations. Briand, Delyon and M´emin in [3] and Ma, Protter, San Mart´ın and Torres in [13] have approximated the Brownian motion by a scaled random walk. Then, in [4], Briand, Delyon and M´emin have studied another case: they approach the Brownian motion by a sequence of martingales.

We are interested in BSDEs with random terminal time because there have strong links with Partial Differ- ential Equations as it is explained by Peng in [14]. As for BSDEs with deterministic terminal time, we study the robustness of numerical methods to approximate the solutions of those BSDEs. In this paper, we shall approximate the Brownian motion either by a scaled random walk, either by a sequence of martingales. In this study, we need moreover to approximate the random almost surely finite terminal timeτ by a sequence of stopping times (τⁿ)_n.

In Section 1, we approximate the Brownian motion by a scaled random walk. First, we shall state the problem and study the properties of existence and uniqueness of the solutions of the BSDEs. Then, we will deal with the convergence of the solutions. To end this part, we give an example of the convergence result for hitting times.

Keywords and phrases.Backward Stochastic Differential Equations (BSDE), stability of BSDEs, weak convergence of filtrations, stopping times.

1 IRMAR, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France;[email protected] c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2006006

In Section 2, we approximate the Brownian motion by a sequence of martingales. We shall see some gen- eralizations of the results of Section 1: existence and uniqueness of the solutions of the BSDEs under study, convergence of the solutions. Moreover, we will illustrate these results by the case of discretizations of a Brow- nian motion and hitting times. For technical reasons, we need some results about convergence of stopped filtrations. So, in Appendix A, we will deal with stopped filtrations and stopped processes. We are going to es- tablish a link between the convergence of a sequence of stopped processes and the convergence of the associated stopped filtrations.

In what follows, we are given a probability space (Ω,A,P). Unless otherwise specified, everyσ-field will be supposed to be included in A, every process will be indexed by R⁺ and taking values in R, every filtration will be indexed by R⁺. D =D(R⁺) denotes the space of c`adl`ag functions fromR⁺ to R. We endow D with the Skorokhod topology. IfX is a process andτ a stopping time, we denote by X^τ the corresponding stopped process,i.e. for everyt,X_t^τ=X_t∧τ.

Definition 0.1. Let (Ft)_t0 be a filtration andτ aF-stopping time. We define theσ-fieldFτ by Fτ ={A∈ F∞:A∩ {τs} ∈ Fs,∀s} where F∞=

t

Ft

and the stopped filtrationF^τ by

F_t^τ=F_τ∧t={A∈ F_t:A∩ {τs} ∈ F_s,∀st}, for everyt.

For technical background about Skorokhod topology, the reader may refer to Billingsley [2] or Jacod and Shiryaev [11].

1. Stability of BSDEs when the Brownian motion is approximated by a scaled random walk

1.1. Statement of the problem

Let f : R² → Rbe a Lipschitz function: there exists K ∈ R⁺ such that, for every (y, z),(y, z) ∈R², we have:

|f(y, z)−f(y, z)|K[|y−y|+|z−z|].

For clarity’s sake, we consider a time-independent generatorf but the results of Section 1 remain true iff is time-dependent as it is explained in Remark 1.3.

We also suppose that f is bounded and that f fills the following property of monotonicity w.r.t. y: there existsµ >0 such that

∀(y, z),(y, z)∈R⁺×R²,(y−y)(f(y, z)−f(y, z))−µ(y−y)².

LetW be a Brownian motion andF its natural filtration. Letτ be aF-stopping time almost surely finite.

We consider the following stochastic differential equation:

Y_t∧τ=ξ+ _τ

t∧τ

f(Y_s, Z_s)ds− _τ

t∧τ

Z_sdW_s, t0, (1)

whereξ is a boundedFτ-mesurable random variable.

Definition 1.1. We call a solution of the BSDE (1) a pair (Y, Z) of progressively measurable processes verifying the equation (1) such thatY_t=ξandZ_t= 0 on the set{t > τ}and:

E

sup

t∈R⁺e^−2µt|Y_t|²

<+∞and∀t, E _t∧τ

0 |Z_t|²dt

<+∞.

According to Theorem 2.1 of Royer in [15], the BSDE (1) has a unique pair solution (Y, Z) in the set of processes such thatY is continuous and uniformly bounded.

In this section, we approximate equation (1) on the following way. We consider the sequence of scaled random walks (Wⁿ)_n1defined by:

W_tⁿ= 1

√n [nt]

k=1

εⁿ_k, t0

where (εⁿ_k)_k∈N∗ is a sequence of i.i.d. symmetric Bernoulli variables. LetFⁿ be the natural filtrations ofWⁿ, n1. We haveF_tⁿ =σ(εⁿ_k, k [nt]). Let (τⁿ)_n be a sequence of bounded (Fⁿ)-stopping times. For eachn, we can findT_n∈Nsuch thatτⁿ T_n.

Then, for eachn, we consider the following equation:

yⁿ_k

n∧τⁿ =yⁿ_k+1

n ∧τⁿ+1

n1_{τn_n^k}f yⁿ_k

n∧τⁿ, zⁿ_k+1

n ∧τⁿ

−zⁿ_k+1

n ∧τⁿ

√1nεⁿ_k+1, k= 0, . . . ,(n−1)T_n

yⁿ_τn =ξⁿ (2)

where (ξⁿ) is a sequence of (F_τⁿn)-measurable integrable random variables.

By a solution of equation (2), we mean a discrete process{yⁿ_k

n, zⁿ_k+1

n }k0that satisfies (2), such thatyⁿ_k

n =ξⁿ andzⁿ_k

n = 0 on the set{τⁿ< ^k_n} and such that{yⁿ_k

n∧τⁿ, zⁿ_k+1

n ∧τⁿ}k0 isF^n,τn-adapted.

Proposition 1.2. Equation (2) has a unique solution (yⁿ, zⁿ).

Proof. We are going to build this solution on a converse iterative way.

Fork=nT_n, let us putyⁿ_k

n∧τⁿ=ξⁿ andzⁿ_k+1

n ∧τⁿ= 0.

Let us suppose that, for a given k, we have built yⁿ_k+1

n ∧τⁿ, zⁿ_k+2

n ∧τⁿ

. Using equation (2), we are going to determinate

yⁿ_k

n∧τⁿ, zⁿ_k+1

n ∧τⁿ

.

Let us begin by giving an expression ofzⁿ_k+1

n ∧τⁿ.

Multiplying equation (2) by√nεⁿ_k+11_{τn^k_n} and taking the conditional expectation with respect toF_k/n^n,τn, we have:

zⁿk+1

n ∧τⁿ1_{τnk n}

= E zⁿ_k+1

n ∧τⁿ1_{τnk

n} F^n,τk ⁿ n

= √nE yⁿ_k+1

n ∧τⁿεⁿ_k+11_{τn^k_n} F^n,τk ⁿ n

+ 1

√nE

1_{τn^k_n}f yⁿ_k

n∧τⁿ, zⁿ_k+1

n ∧τⁿ

εⁿ_k+1 F^n,τk ⁿ n

−√nE yⁿ_k

n∧τⁿεⁿ_k+11_{τn_n^k} F^n,τk ⁿ n

= √nE yⁿ_k+1

n ∧τⁿεⁿ_k+1 F^n,τk ⁿ n

1_{τnk

n}

becauseyⁿ_k

n∧τⁿ andzⁿ_k+1

n ∧τⁿ must beF^n,τk ⁿ

n -measurable,τⁿis aFⁿ-stopping time andεⁿ_k+1is independent from F^n,τk ⁿ

n and is centered.

Moreover, by definition of a solution, for everyk,zⁿ_k+1

n ∧τⁿ=zⁿ_k+1

n ∧τⁿ1_{τnk

n}. So, we put:

zⁿ_k+1

n ∧τⁿ=√ nE

yⁿ_k+1

n ∧τⁿεⁿ_k+1 Fⁿk n

1_{τn_n^k}. We point out thatzⁿ_k+1

n ∧τⁿ isF^n,τk ⁿ

n -measurable.

Now, we have to determinateyⁿ_k

n∧τⁿ. On the set{τⁿ< ^k_n},yⁿ_k

n∧τⁿ=yⁿ_k+1

n ∧τⁿ=y_τⁿn=ξⁿ. On{τ^{n k}_n},yⁿ_k

n∧τⁿ=yⁿ_k+1

n ∧τⁿ+_n¹f yⁿ_k

n∧τⁿ, zⁿ_k+1

n ∧τⁿ

−zⁿ_k+1

n ∧τⁿ√1

nεⁿ_k+1 and we can write it yⁿk

n∧τⁿ=ϕ yⁿk

n∧τⁿ

withϕ(y) =yⁿ_k+1

n ∧τⁿ+¹_nf y, zⁿ_k+1

n ∧τⁿ

−zⁿ_k+1

n ∧τⁿ√1

nεⁿ_k+1. Asf isK-Lipschitz iny, we have, for everyy, y:

|ϕ(y)−ϕ(y)| K

n|y−y|.

So, for n large enough (notice that this range does not depend onk), ^K_n <1 and ϕis a contraction. Then, the equation yⁿ_k

n∧τⁿ =ϕ yⁿ_k

n∧τⁿ

has a unique solution forn large enough, according to a fix point Theorem.

By construction, yⁿ_k

n∧τⁿ is F^n,τk+1ⁿ

n -measurable. But, using the predictable representation property, we have yⁿ_k+1

n ∧τⁿ−zⁿ_k+1

n ∧τⁿ√1nεⁿ_k+1 = E[yⁿk+1

n ∧τⁿ|F^n,τk ⁿ

n ]. So yⁿ_k

n∧τⁿ is independant from εⁿ_k+1 and yⁿ_k

n∧τⁿ is F^n,τk ⁿ n - measurable.

Hence, the equation (2) has a unique solution.

Now, we define continuous time processes Yⁿ and Zⁿ byY_tⁿ =yⁿ_[nt]

n ∧τⁿ, Z_tⁿ =zⁿ_nt

n ∧τⁿ, for everyt ∈R⁺ where x= (x−1)⁺ ifxis an integer, [x] otherwise. The processesYⁿ andZⁿ are constant on the intervals [k/n,(k+ 1)/n[ and ]k/n,(k+ 1)/n] respectively and satisfy the following equation:

Y_tⁿ=ξⁿ+ _τⁿ

t∧τⁿ

f(Y_(s∧τⁿ n)−, Z_s∧τⁿ n)dAⁿ_s− _τⁿ

t∧τⁿ

Z_s∧τⁿ ndW_sⁿ, (3)

whereAⁿ_s =^[ns]_n .

Remark 1.3. Iff is time-dependent such that for every (y, z), we suppose that{f(t, y, z)}t0is progressively measurable, bounded, K-Lipschitz in y andz and verify the condition of monotonicity w.r.t. y given before.

Under these assumptions, all the results of Section 1 remain true and the proofs are the same. In that case, Equation (2) becomes:

yⁿ_k

n∧τⁿ = yⁿ_k+1

n ∧τⁿ+1

n1_{τn^k_n}f k

n∧τⁿ, yⁿ_k

n∧τⁿ, zⁿ_k+1

n ∧τⁿ

−zⁿ_k+1

n ∧τⁿ

√1nεⁿ_k+1, k= 0, . . . ,(n−1)T_n yⁿ_τn = ξⁿ.

1.2. Convergence of the solutions

The aim of this section is to prove the following result of convergence of the solutions:

Theorem 1.4. Let (Y, Z) be the solution of the BSDE (1) and (Yⁿ, Zⁿ) be the processes constant on the intervals [k/n,(k+ 1)/n[ and ]k/n,(k+ 1)/n] respectively solving equation (3). We suppose that there exists δ > 0 such that sup_nE[|ξⁿ|^1+δ]^1+δ1 <+∞ and sup_nE[|τⁿ|^1+δ]^1+δ1 <+∞ and that we have the convergences ξⁿ −→^P ξ,τⁿ−→^P τ andWⁿ−→^P W. Then

∀L∈N, sup

t∈[0,L]|Y_t∧τⁿ n−Y_t∧τ|+ _τ∧τⁿ

0 |Z_t∧τⁿ n−Z_t∧τ|²dt−→^P 0 which we shall denote by (Yⁿ, Zⁿ)→(Y, Z)and also

∀L∈N, sup

t∈[0,L]

_t∧τⁿ

0

Z_sⁿdW_sⁿ− _t∧τ

0

Z_sdW_s −→^P 0.

According to this theorem, we have the convergence in probability of the solutions under the rather strong assumption that the scaled random walks converge in probability to the Brownian motion and the random terminal times also converge. Actually, with Donsker’s Theorem, we have the convergence in law of the scaled random walks to the Brownian motion. If we only have this convergence in law, we obtain the following corollary:

Corollary 1.5. Let (Y, Z) be the solution of the BSDE (1) and (Yⁿ, Zⁿ) be the processes constant on the intervals [k/n,(k+ 1)/n[ and ]k/n,(k + 1)/n] respectively solution of the equation (3). We suppose that

∀n,∀k, εⁿ_k = ε_k, ξ = g(W) and ξⁿ = g(Wⁿ) with g bounded continuous. We assume that there exists δ > 0 such that sup_nE[|τⁿ|^1+δ]^1+δ1 <+∞. We also suppose that we have the convergence (Wⁿ, τⁿ)−→^L (W, τ). Then Y_.∧τⁿ n,_.∧τⁿ

0 Z_sⁿdW_sⁿ

n converges in law to

Y_.∧τ,_.∧τ

0 Z_sdW_s

for the Skorokhod topology.

Proof. According to the Skorokhod representation theorem, in a space ( ˜Ω,A˜,˜P), we can find ( ˜Wⁿ,τ˜ⁿ) with the same law as (Wⁿ, τⁿ) and ( ˜W ,τ) with the same law as (W, τ˜ ) such that ( ˜Wⁿ,τ˜ⁿ)−−→^a.s. ( ˜W ,τ). We denote by˜ F˜ the natural filtration of ˜W and by ˜Fⁿ the natural filtrations of the ˜Wⁿ.

Let us show that ˜τ is an ˜F-stopping time. Let us fix t0 and note that{τ t} ∈ Ftsinceτ is aF-stopping time. Then, for everyε >0, we can findh:R^k →Rmeasurable ands₁, . . . , s_k in [0, t] such that

|1_{τt}−h(W_s1, . . . , W_sk)|dP< ε.

(W, τ)∼( ˜W ,τ) so˜ 1_{τt}−h(W_s1, . . . , W_sk)∼1_{˜τt}−h( ˜W_s1, . . . ,W˜_sk). Then,

|1_{˜τt}−h( ˜W_s1, . . . ,W˜_sk)|d˜P=

|1_{τt}−h(W_s1, . . . , W_sk)|dP. So, for everyε >0, we can findh:R^k →Rmeasurable and s₁, . . . , s_k in [0, t] such that

|1_{˜τt}−h( ˜W_s1, . . . ,W˜_sk)|d˜P< ε.

Hence,{τ˜t} ∈F˜t. Then ˜τ is an ˜F-stopping time. On the same way, the ˜τⁿ’s are ˜Fⁿ-stopping times.

Let (Yⁿ, Zⁿ) be the solution of Y_tⁿ=g( ˜Wⁿ) +

_τ˜ⁿ

t∧˜τⁿ

f(Y_(s∧˜ⁿ_τn)−, Z_s∧˜ⁿ_τn)dAⁿ_s − _˜τⁿ

t∧˜τⁿ

Z_s∧˜ⁿ_τnd ˜W_sⁿ

and (Y, Z) be the solution of

Y_{t∧˜ τ} =g( ˜W) + _τ˜

t∧˜τ

f(Y_s, Z_s)ds− _˜τ

t∧˜τ

Z_sd ˜W_s, t0.

All the assumptions of Theorem 1.4 are filled. So we have the convergence in probability of

Y_.∧˜ⁿ_τn,_.∧˜τⁿ

0 Z_sⁿd ˜W_sⁿ

n

to

Y_{.∧˜ τ},_.∧˜τ

0 Z_sd ˜W_s

. Then, denoting by (Y_.∧τⁿ n, Z_.∧τⁿ n) the solution of (1) and by (Y_.∧τ, Z_.∧τ) the solution of (3), since (Y_.∧τⁿ n, Z_.∧τⁿ n, Wⁿ, τⁿ) has the same law as (Y_.∧˜ⁿ_τn, Z_.∧˜ⁿ_τn,W˜ⁿ,τ˜ⁿ) and (Y_.∧τ, Z_.∧τ, W, τ) has the same law as (Y_.∧˜τ, Z_{.∧˜ τ},W ,˜ ˜τ), we have the convergence of

Y_.∧τⁿ n,_.∧τⁿ

0 Z_sⁿdW_sⁿ

n to

Y_.∧τ,_.∧τ

0 Z_sdW_s in

distribution for the Skorokhod topology.

We point out that in this corollary, it is necessary to have the joint convergence of ((Wⁿ, τⁿ))_nto (W, τ). In section 1.3, we shall see an example where the stopping times are hitting times.

To prove the first convergence of Theorem 1.4, we are going to use the Picard approximations (Y^p, Z^p) and ((Y^n,p, Z^n,p))_n defined on the following way:

Y_t∧τ^p+1 = ξ+ _τ

t∧τ

f(Y_s^p, Z_s^p)ds− _τ

t∧τ

Z_s^p+1dW_s, ∀t0, (4)

Y_t^n,p+1 = ξⁿ+ _τⁿ

t∧τⁿ

f(Y_s−^n,p, Z_s^n,p)dAⁿ_s− _τⁿ

t∧τⁿ

Z_s^n,p+1dW_sⁿ, (5)

withY⁰=Z⁰=Y^n,0=Z^n,0= 0 and Aⁿ_t = ^[nt]_n .

We have existence and uniqueness of adapted pairs (Y^p, Z^p) and ((Y^n,p, Z^n,p))_n by induction on p, using respectively Theorem 2.1 of Royer in [15] and Proposition 1.2.

We write:

Yⁿ−Y = (Yⁿ−Y^n,p) + (Y^n,p−Y^p) + (Y^p−Y), Zⁿ−Z = (Zⁿ−Z^n,p) + (Z^n,p−Z^p) + (Z^p−Z).

We are going to prove successively the convergence of the Picard approximations to the solutions, i.e. for every n, (Y^n,p, Z^n,p) → (Yⁿ, Zⁿ) and (Y^p, Z^p) → (Y, Z), and then check that the first convergence of the theorem is true for the Picard approximations,i.e. for everyp, (Y^n,p, Z^n,p)→(Y^p, Z^p).

Most of the proof uses the same arguments as in the proof of Theorem 2.1 of Briand, Delyon and M´emin in [3] but there are some technical difficulties due to the stopping times.

The arguments of Lemma 4.1 in [3] are still true when we replace a deterministic terminal time by a bounded random terminal time. So we have:

∀L, sup

n E

sup

t∈[0,L]|Y_τⁿn∧t−Y_τ^n,pn∧t|²+ _+∞

0 |Z_tⁿ−Z_t^n,p|²dt

−−−−−→

p→+∞ 0. (6)

With a truncation argument (using the fact thatτis almost surely finite), we deduce the convergence of (Y^p, Z^p) to (Y, Z),i.e.

∀L, E

t∈[0,L]sup |Y_τ∧t^p −Y_τ∧t|²+ _+∞

0 |Z_t^p−Z_t|²dt

−−−−−→

p→+∞ 0. (7)

Now, let us show that for everyp, (Y^n,p, Z^n,p)→(Y^p, Z^p) as n→+∞that is

∀L∈N, sup

t∈[0,L]|Y_t∧τ^n,pn−Y_t∧τ^p |+ _τⁿ

0 |Z_t∧τ^n,pn−Z_t∧τ^p |²dt−→^P 0 asn→ ∞. (8) We argue by induction on p.

– The property is true forp= 0 because Y⁰=Z⁰=Y^n,0=Z^n,0= 0.

– We suppose that, for given p, (8) holds. Let us prove that (8) is still true for p+ 1. The proof will be given through three steps.

In a first step, we introduce the sequence (Mⁿ)_n of processes defined by M_tⁿ=Y_t∧τ^n,p+1n +

_t∧τⁿ

0

f(Y_s^n,p, Z_s^n,p)dAⁿ_s and the processM defined by

M_t=Y_t∧τ^p+1+ _t∧τ

0

f(Y_s^p, Z_s^p)ds.

In Lemma 1.6, we prove that (M_.∧τⁿ n)_n is a sequence of (F^n,τn)-martingales. Then, with Lemmas 1.8 and 1.9, we show that we have the following convergence:

∀L, sup

t∈[0,L]|M_t∧τⁿ n−M_t∧τ|−→^P 0 asn→ ∞. The aim of the second step is to prove Lemma 1.11:

_τⁿ

0 |Z_t∧τ^n,p+1n −Z_t∧τ^p+1|²dt−→^P 0 asn→ ∞.

At last, in a third step (Lemma 1.12), we deal with the convergence ofY^n,p+1 toY^p+1:

∀L, sup

t∈[0,L]|Y_t∧τ^n,p+1n −Y_t∧τ^p+1|−→^P 0 asn→ ∞. Step 1. Let (Mⁿ)_n be the processes defined byM_tⁿ =Y_t∧τ^n,p+1n +_t∧τⁿ

0 f(Y_s^n,p, Z_s^n,p)dAⁿ_s. Lemma 1.6. For each n,Mⁿ is aF^n,τn-martingale.

Proof. Let us show thatM_tⁿ=M₀ⁿ+_t∧τⁿ

0 Z_s∧τ^n,p+1n dW_sⁿ. Using (5), M₀ⁿ+

_t∧τⁿ

0

Z_s∧τ^n,p+1n dW_sⁿ

= Y₀^n,p+1+ _t∧τⁿ

0

Z_s∧τ^n,p+1n dW_sⁿ

= ξⁿ+ _τⁿ

0

f(Y_s−^n,p, Z_s^n,p)dAⁿ_s − _τⁿ

0

Z_s^n,p+1dW_sⁿ+ _t∧τⁿ

0

Z_s∧τ^n,p+1n dW_sⁿ

= Y_t∧τ^n,p+1n + _t∧τⁿ

0

f(Y_s−^n,p, Z_s^n,p)dAⁿ_s =M_tⁿ.

Then, for every n, as the process (M₀ⁿ+_t

0Z_s∧τ^n,p+1n dW_sⁿ)_t0 is a Fⁿ-martingale, the stopped process (M₀ⁿ+ _t∧τⁿ

0 Z_s∧τ^n,p+1n dW_sⁿ)_t0= (M_tⁿ)_t0 is aF^n,τn-martingale.

So, we have M_tⁿ=E[M_τⁿn|F_t^n,τn] withM_τⁿn=Y_τ^n,p+1n +_τⁿ

0 f(Y_s−^n,p, Z_s^n,p)dAⁿ_s, for everyn, for everyt.

Before proving the convergence of (M_τⁿn)_n, let us see a lemma that links stopped integrals and integrals of stopped processes:

Lemma 1.7. For every t, we have the following relations:

_t∧τ

0

f(Y_s^p, Z_s^p)ds= _t

0

f(Y_s∧τ^p , Z_s∧τ^p )ds+ (τ∧t−t)f(Y_τ^p, Z_τ^p), _t∧τⁿ

0

f(Y_s^n,p, Z_s^n,p)ds= _t

0

f(Y_s∧τ^n,pn, Z_s∧τ^n,pn)ds+ (τⁿ∧t−t)f(Y_τ^n,pn , Z_τ^n,pn).

Proof. Let us prove the first relation.

On{tτ}, _t

0

f(Y_s∧τ^p , Z_s∧τ^p )ds= _t

0

f(Y_s^p, Z_s^p)ds= _t∧τ

0

f(Y_s^p, Z_s^p)ds and on{t > τ}, we have:

_t

0 f(Y_s∧τ^p , Z_s∧τ^p )ds = _t∧τ

0 f(Y_s∧τ^p , Z_s∧τ^p )ds+ _t

t∧τf(Y_s∧τ^p , Z_s∧τ^p )ds

= _t∧τ

0

f(Y_s^p, Z_s^p)ds+ _t

t∧τ

f(Y_τ^p, Z_τ^p)ds

= _t∧τ

0

f(Y_s^p, Z_s^p)ds+ (t−t∧τ)f(Y_τ^p, Z_τ^p).

The second equality is proved by the same arguments.

Lemma 1.8. (M_τⁿn)_n converges in L¹ toY_τ^p+1+_τ

0 f(Y_s^p, Z_s^p)ds.

Proof. It suffices to show that (M_τⁿn)_n converges toY_τ^p+1+_τ

0 f(Y_s^p, Z_s^p)dsin probability. Indeed, for everyn, E[|M_τⁿn|^1+δ]^1+δ1 E[|ξⁿ|^1+δ]^1+δ1 +f∞E[|τⁿ|^1+δ]^1+δ1 . So, when we take the sup inn, sup_nE[|M_τⁿn|^1+δ]^1+δ1 <∞ according to the assumptions on (ξⁿ)_n, (τⁿ)_n and f. So, the sequence (M_τⁿn) is uniformly integrable and then the convergence in probability implies the convergence in L¹.

Let us show that (M_τⁿn)_n converges in probability toY_τ^p+1+_τ

0 f(Y_s^p, Z_s^p)ds.

M_τⁿn−Y_τ^p+1+ _τ

0 f(Y_s^p, Z_s^p)ds

|Y_τ^n,p+1n −Y_τ^p+1|+ _τⁿ

0 f(Y_s−^n,p, Z_s^n,p)dAⁿ_s− _τ

0 f(Y_s^p, Z_s^p)ds . But,

|Y_τ^n,p+1n −Y_τ^p+1|=|ξⁿ−ξ|−→^P 0. (9) We are going to conclude by showing that

_τⁿ

0

f(Y_s−^n,p, Z_s^n,p)dAⁿ_s − _τ

0

f(Y_s^p, Z_s^p)ds −→^P 0.

For eachn, for eachω, there exists a uniquek_nsuch thatτⁿis in the interval [k_n/n,(k_n+1)/n[. As the processes Y^n,p and Z^n,p are constant on the intervals of the form [k/n,(k+ 1)/n[ and ]k/n,(k+ 1)/n] respectively by