In [5] the authors put some constrains on the solution of BSDE

doi:10.7151/dmps.1145

DISCRETE APPROXIMATIONS OF GENERALIZED RBSDE WITH RANDOM TERMINAL TIME

Katarzyna Ja´nczak-Borkowska Institute of Mathematics And Physics University of Technology and Life Sciences

Kaliskiego 7, 85–796 Bydgoszcz, Poland e-mail: kaja@utp.edu.pl

Abstract

The convergence of discrete approximations of generalized reflected back- ward stochastic differential equations with random terminal time in a gen- eral convex domain is studied. Applications to investigation obstacle elliptic problem with Neumann boundary condition for partial differential equations are given.

Keywords: generalized reflected BSDE, discrete approximation methods, viscosity solution.

2010 Mathematics Subject Classification: Primary: 60H10, 60Gxx;

Secondary: 35J20.

1. Introduction

Pardoux and Peng in [13] have introduced nonlinear backward stochastic differen- tial equations (BSDEs for short). Since then many papers have been devoted to the study of BSDEs, mainly due to their applications. The main aim of studying BSDEs was to give a probabilistic interpretation for solutions of partial differen- tial equations (PDEs for short).

In [5] the authors put some constrains on the solution of BSDE. They assumed that the first component of the solution takes its values in a given convex set D ⊂ R^d and the problem was called the reflected BSDE (RBSDE for short). Later, Pardoux and R˘a¸scanu in [14] considered RBSDE with random terminal time and pointed out the connection between RBSDEs and variational inequalities.

The notion of generalized BSDEs was introduced in [15]. By generalized it is meant that to the stochastic equation an additional component – an integral with respect to one dimensional increasing process – is added. Recently papers about the generalized RBSDEs on a finite time interval have appeared. The paper [16] treats one dimensional case and [7] treats d-dimensional case. The existence and uniqueness of the generalized RBSDE (GRBSDE for short) with random terminal time and its connection with PDEs with an obstacle problem and Neumann boundary condition was shown in [8].

In the literature we can find many papers related to discrete approximations of backward stochastic equations. We should list here [1, 3, 4, 10] that treat the case of BSDEs with deterministic terminal time and [18] that treats the case of BSDEs with random terminal time. Discrete approximations of RBSDEs firstly were proposed in one-dimensional case (e.g. [1, 11]) and later also in d- dimensional case (see [6]). In [7] an approximation scheme for the solution of generalized RBSDE with deterministic terminal time was given.

In the present paper we propose the discrete approximation of the solution of the following GRBSDE with random terminal time τ :

Y_t∧τ = ξ + Z τ

t∧τ

f (s, Y_s, Z_s)ds + Z τ

t∧τ

ϕ(s, Y_s)dΛ_s (1)

− Z τ

t∧τ

ZsdWs+ Kτ − K_t∧τ, t ∈ R⁺,

where W = (Wt)_t∈R⁺ is m-dimensional Wiener process and Λ = (Λt)_t∈R⁺ is one dimensional continuous and increasing process, Λ0 = 0. Moreover, we show the convergence of the proposed scheme to the solution of (1).

The paper is organized as follows. In Section 2 we give a definition of a solution of GRBSDE with random terminal time. We formulate here a theorem about existence and uniqueness of the solution of (1). In the next section we construct a discrete approximation scheme for solving (1) and give its properties.

Moreover, this section is devoted to proving the main theorem - theorem about convergence of the proposed scheme. Finally, in the last section we show the application of the constructed scheme to numerical solving of the obstacle problem for PDE with Neumann boundary condition.

Throughout the paper we will use the following notations. By |x| we mean an Euclidean norm in R^d, x ∈ R^d, kxk stands for (trace(x^∗x))^1/2, where x^∗ is a transposition of a matrix x ∈ R^d×m. For a process K = (K¹, ..., K^d) by

|K|_t=Pd

i=1|Kⁱ|_twe denote its variation on [0, t], where |Kⁱ|_tis a total variation of Kⁱ on [0, t].

2. Definition

Let (Ω, G, P) be a complete probability space carrying a standard m-dimensional Wiener process W = (Wt)_t∈R⁺. Let F = (Ft)_t∈R⁺ be the usual augmentation of the filtration generated by W and assume that Λ = (Λ_t)_t∈R⁺ is an adapted, one dimensional continuous and increasing process, Λ0 = 0.

Let τ be an almost surely finite F stopping time and let ξ be an F_τ measur- able, square integrable random variable with values in ¯D, where D is a convex subset of R^d. Suppose that functions f : R⁺ × Ω × R^d× R^d×m → R^d and ϕ : R⁺× Ω × R^d→ R^d are measurable.

Definition. A solution of the generalized reflected backward stochastic differ- ential equation (GRBSDE) with random terminal time associated with data (τ, ξ, f, ϕ, Λ) is a triple (Y, Z, K) = (Yt, Zt, Kt)_t∈R⁺ of F progressively measurable processes in ¯D × R^d×m× R^d satisfying

Y_t = ξ + Z τ

t∧τ

f (s, Y_s, Z_s)ds + Z τ

t∧τ

ϕ(s, Y_s)dΛ_s (2)

− Z τ

t∧τ

Z_sdW_s+ K_τ − K_t∧τ, t ∈ R⁺ and such that for some λ ∈ R

E

 sup

t≤τ

e^λt|Y_t|²+ Z τ

0

e^λtkZ_tk²dt + Z τ

0

e^λt|Y_t|²dΛt



< ∞, where K is a continuous process with locally finite variation, K0 = 0 and

Z τ 0

(Y_t− S_t)dK_t≤ 0, (3)

for every F progressively measurable process S = (S_t)_t∈R+ with values in ¯D.

Moreover, on the set {t ≥ τ } we have Yt= ξ, Zt= 0, Kt= Kτ.

Assume that there exist constants L, κ > 0, β < 0, µ ∈ R such that for any t ∈ R⁺, y, y⁰ ∈ R^d, z, z⁰ ∈ R^d×m

(A1) • |f (t, y, z) − f (t, y⁰, z⁰)| ≤ L(|y − y⁰| + kz − z⁰k),

• hy − y⁰, f (t, y, z) − f (t, y⁰, z)i ≤ µ|y − y⁰|², (A2) • |ϕ(t, y) − ϕ(t, y⁰)| ≤ L|y − y⁰|,

• hy − y⁰, ϕ(t, y) − ϕ(t, y⁰)i ≤ β|y − y⁰|²,

• |ϕ(·, ·, ·)| ≤ κ,

(A3) for some real numbers λ and ν such that λ > 2µ + L², ν > β E

 Z τ 0

e^λt+νΛt|f (t, 0, 0)|²dt + Z τ

0

e^λt+νΛt|ϕ(t, 0)|²dΛt



< ∞,

(A4) ξ is F_τ measurable random variable with values in D,¯ Ee^{λτ +νΛτ}(|ξ|²+ 1) < ∞ and

E

 Z τ 0

e^λt+νΛt|f (t, ξ_t, ζ_t)|²dt + Z τ

0

e^λt+νΛt|ϕ(t, ξ_t)|²dΛ_t



< ∞,

where ξ_t = E(ξ|F_t), ζ is F progressively measurable d × m-dimensional process such that ER∞

0 kζ_tk²dt < ∞ and ξ = Eξ +R∞ 0 ζtdWt, (A5) there exists q ≥ 2 such that for every M > 0

E Z M

0

|f (t, 0, 0)|^2qdt < ∞.

Theorem 1 [8, Theorem 2.2 and Proposition 4.1]. Let τ be an almost surely finite F stopping time and let assumptions (A1)–(A5) hold. Then there exists a unique solution of (2).

Moreover, for any a ∈ D there exists C > 0 such that for λ > 2µ + L² E

 sup

t≤τ

e^λt|Y_t− a|²+ Z τ

0

e^λt|Y_t− a|²dΛt+ Z τ

0

e^λtkZ_tk²dt + Z τ

0

e^λtd|K|t



≤ CE



e^λτ|ξ − a|²+ Z τ

0

e^λt|f (t, a, 0)|²dt + Z τ

0

e^λt|ϕ(t, a)|²dΛ_t

 .

3. Convergence of the approximation scheme

We will give an approximation scheme for (2) based on an approximation of a Wiener process by a random walk. In a construction of this scheme and in the proof of its convergence we will use results from [7], where a scheme for (2) when τ is a deterministic terminal time was given.

Set W_tⁿ = (√

n)⁻¹P[nt]

j=1εⁿ_j, t ∈ R⁺, where for each n ∈ N, {εⁿj}_j∈N is a sequence of independent symmetric Bernoulli random variables, and by Fⁿ = {F_tⁿ}_t∈R+ denote the natural filtration of Wⁿ.

By Λ^n,L we will denote a process with bounded jumps such that Λ^n,L_t = P[nt]

j=1∆Λ^n,L_(j−1)/n, where ∆Λ^n,L_j/n = min((2L)⁻¹, ∆Λⁿ_j/n) and ∆Λⁿ_j/n = Λⁿ_j/n −

Λⁿ_(j−1)/n. Note that since Λ has continuous trajectories, max_j≤[nTn]|∆Λⁿ_j/n| → 0 and sup_t≤Tn|Λ^n,L_t − Λ_t| ≤P[nT ]

j=1 |1/(2L) − ∆Λⁿ_j/n|1_{∆Λn

j/n>1/2L} → 0, a.s.

Now we need to approximate the stopping time τ by a sequence of bounded stopping times {τⁿ}_n. Since for every n ∈ N, τⁿ is bounded, we can find Tn∈ N such that τⁿ≤ T_n. Denote τ_jⁿ= j/n ∧ [nτⁿ]/n, j ∈ N.

In order to define the discrete GRBSDE with random terminal time we take j = nT_n, . . . , 0 and on the set {τⁿ ≤ j/n} put y_τⁿn

j = yⁿ_τn = ξⁿ = E(ξ|F_τⁿn) and zⁿ_τn

j = z_τⁿn = 0, ∆k_τⁿn

j+1 = 0. Next, on the set {[nτⁿ] > j} we solve y_j/nⁿ = y_τⁿⁿ

j+1+ 1

nf (j/n, yⁿ_j/n, z_j/nⁿ )1_{[nτⁿ_]>j}

(4)

+ ϕ(j/n, yⁿ_j/n)1_{[nτⁿ_]>j}∆Λ^n,L_j/n− 1

√nz_j/nⁿ εⁿ_j+1+ ∆k_τⁿⁿ

j+1.

By a solution of (4) we mean a triple (Yⁿ, Zⁿ, Kⁿ) = (Y_tⁿ, Z_tⁿ, K_tⁿ)_t∈[0,Tn] of Fⁿ adapted processes in ¯D×R^d×m×R^dsuch that K₀ⁿ= 0 andRτⁿ

0 (Y_t−ⁿ−S_t−ⁿ )dK_tⁿ≤ 0 for every Fⁿ adapted process Sⁿwith values in ¯D, where Y_tⁿ= yⁿ_τⁿ

[nt], Z_tⁿ= z_τⁿⁿ

[nt], K_tⁿ = P[nt]

j=1∆k_τⁿⁿ

j. Moreover, on the set {t ≥ τⁿ}, Y_tⁿ = Y_τⁿⁿ, Z_tⁿ = 0 and K_tⁿ= K_τⁿⁿ.

Theorem 2. Assume (A1)–(A5). Let {τⁿ} be a sequence of Fⁿ stopping times such that sup_nEe^λτn(|ξⁿ|²+ 1) < ∞ and τⁿ−→ τ . Then for every T ∈ R^{P +}

n→∞lim E

 sup

t≤T

|Y_t∧τⁿ n − Y_t∧τ|²+ Z T

0

kZ_t−^n,τn− Z_t^τk²dt + Z T

0

|Y_t−^n,τn− Y_t^τ|dΛ_t + sup

t≤T

|K_t∧τⁿ n− K_t∧τ|²



= 0.

Before proving the above theorem we will give some properties of the discrete scheme. First note, that the triple (Yⁿ, Zⁿ, Kⁿ) is a unique solution of (4), that is equivalent to

Y_t∧τⁿ n = ξⁿ+ Z τⁿ

t∧τⁿ

f (%ⁿ_s−, Y_s−ⁿ, Z_s−ⁿ )d%ⁿ_s + Z τⁿ

t∧τⁿ

ϕ(%ⁿ_s−, Y_s−ⁿ)dΛ^n,L_s (5)

− Z τⁿ

t∧τⁿ

Z_s−ⁿ dW_sⁿ+ K_τⁿn− K_t∧τⁿ n, t ∈ R⁺,

where %ⁿ_t = [nt]/n. Now, note that solving (4) relies on finding on the set {[nτⁿ] > j}

z_j/nⁿ = z_j/nⁿ 1_{[nτⁿ_]>j}=√

nE(yⁿ_τⁿ

j+1εⁿ_j+1|F_j/nⁿ )1_{[nτⁿ_]>j}, hⁿ_j/n = E(y_τⁿⁿ

j+1|F_j/nⁿ ) + 1

nf (j/n, π(hⁿ_j/n), zⁿ_j/n)1_{[nτⁿ_]>j}

+ ϕ(j/n, π(hⁿ_j/n))1{[nτⁿ]>j}∆Λ^n,L_j/n, y_j/nⁿ = π(hⁿ_j/n),

∆kⁿ_τⁿ

j+1 = y_j/nⁿ − hⁿ_j/n.

Since f and ϕ are Lipschitz, hⁿ_j/nis well defined for n > 2L . Moreover processes defined above satisfy

Z T 0

(Y_t−ⁿ − S_t−ⁿ )dK_tⁿ≤ 0.

(6)

Indeed, since D is a convex set hπ(h)−x⁰, π(h)−hi ≤ 0, h ∈ R^d, x⁰ ∈ ¯D (see ex.

[12]). In particular, hy_(j−1)/nⁿ − x⁰, ∆kⁿ_j/ni ≤ 0 for any x⁰ ∈ ¯D and j = 1, . . . , [nT_n] and as a consequence, for any Fⁿ adapted process Sⁿ with values in ¯D

Z T 0

(Y_t−ⁿ − S_t−ⁿ )dK_tⁿ=

[nT ]

X

j=1

hy_(j−1)/nⁿ − S_(j−1)/nⁿ , ∆k_j/nⁿ i ≤ 0.

It can be shown that E sup_t≤τⁿe^λt|Y_tⁿ|² < ∞ and for t ∈ R⁺, E(kZ_tⁿk²+ |Kⁿ|²_t)

< ∞.

Proposition 3. Assume that (A1)–(A4) are satisfied and τⁿ is a sequence of Fⁿ stopping times such that sup_nEe^λτn(1 + |ξⁿ|²) < ∞.

(a) There exists a constant C > 0 such that for any n ∈ N and a ∈ D

E

 sup

t≤τⁿ

e^λt|Y_tⁿ− a|²+ Z τⁿ

0

e^λ%ntkZ_t−ⁿ k²d%ⁿ_t + Z τⁿ

0

e^λ%nt|Y_t−ⁿ − a|²dΛ^n,L_t



≤ CE



e^λτn|ξⁿ− a|²+ Z τⁿ

0

e^λ%nt|f (%ⁿ_t−, a, 0)|²d%ⁿ_t + Z τⁿ

0

e^λ%nt|ϕ(%ⁿ_t−, a)|²dΛ^n,L_t



and

E

 sup

t≤τⁿ

e^λt|K_tⁿ|²+ e^λτn|Kⁿ|²_τn



≤ CE



1 + e^λτn|ξⁿ− a|² Z τⁿ

0

e^λ%nt|f (%ⁿ_t−, a, 0)|²d%ⁿ_t + Z τⁿ

0

e^λ%nt|ϕ(%ⁿ_t−, a)|²dΛ^n,L_t

 .

(b) The solution of (5) is unique.

Proof. (a) By the Itˆo formula

e^λ(t∧τn) |Y_t∧τⁿ n− a|²+ λ Z τⁿ

t∧τⁿ

e^λs|Y_s−ⁿ − a|²d%ⁿ_s + Z τⁿ

t∧τⁿ

e^λ%nsd[Yⁿ]s

= e^λτn|ξⁿ− a|²+ 2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)f (%ⁿ_s−, Y_s−ⁿ, Z_s−ⁿ )d%ⁿ_s

+2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)ϕ(%ⁿ_s−, Y_s−ⁿ)dΛ^n,L_s + 2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)dK_sⁿ

−2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)Z_s−ⁿ dW_sⁿ.

Since exponent function is nonnegative and by (6), Rτⁿ

t∧τⁿe^λ%ns(Y_s−ⁿ − a)dK_sⁿ≤ 0.

Using assumptions on f and ϕ and by the equality [Yⁿ]t=Rt

0kZ_s−ⁿ k²d%ⁿ_s e^λ(t∧τn) |Y_t∧τⁿ n− a|²+ λ

Z τⁿ t∧τⁿ

e^λ%ns|Y_s−ⁿ − a|²d%ⁿ_s + Z τⁿ

t∧τⁿ

e^λ%nskZ_s−ⁿ k²d%ⁿ_s

≤ e^λτn|ξⁿ− a|²+ (2µ + L²/ε + η) Z τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ − a|²d%ⁿ_s

+ ε Z τⁿ

t∧τⁿ

e^λ%nskZ_s−ⁿ k²d%ⁿ_s + 1/η Z τⁿ

t∧τⁿ

e^λ%ns|f (%ⁿ_s−, a, 0)|²d%ⁿ_s

+β Z τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ − a|²dΛ^n,L_s + 1/|β|

Z τⁿ t∧τⁿ

e^λ%ns|ϕ(%ⁿ_s−, a)|²dΛ^n,L_s

−2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)Z_s−ⁿ dW_sⁿ.

Let ε, η > 0 be such that ˜ε = 1 − ε > 0 and ˜λ = λ − (2µ + L²/ε + η) > 0. Since β < 0,

e^λ(t∧τn) |Y_t∧τⁿ n− a|²+ ˜λ Z τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ − a|²d%ⁿ_s (7)

+ |β|

Z τⁿ t∧τⁿ

e^λ%ns|Y_s−ⁿ − a|²dΛ^n,L_s + ˜ε Z τⁿ

t∧τⁿ

e^λ%nskZ_s−ⁿ k²d%ⁿ_s

≤ e^λτn|ξⁿ− a|²+ C Z τⁿ

t∧τⁿ

e^λ%ns|f (%ⁿ_s−, a, 0)|²d%ⁿ_s

+ C Z τⁿ

t∧τⁿ

e^λ%ns|ϕ(%ⁿ_s−, a)|²dΛ^n,L_s − 2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)Z_s−ⁿ dW_sⁿ, where by C we denoted a constant which values may vary from line to line. Since {R·

0e^λs(Y_s − a)Z_sdW_s} is a martingale, after integrating the above inequality we get

E



e^λ(t∧τn)|Y_t∧τⁿ n− a|² + Z _τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ − a|²dΛ^n,L_s + Z _τⁿ

t∧τⁿ

e^λ%nskZ_s−ⁿ k²d%ⁿ_s

 (8)

≤ CE



e^λτn|ξⁿ− a|²+ Z τⁿ

t∧τⁿ

e^λ%ns|f (%ⁿ_s−, a, 0)|²d%ⁿ_s

+ Z τⁿ

t∧τⁿ

e^λ%ns|ϕ(%ⁿ_s−, a)|²dΛ^n,L_s

 . Note that

E sup

t≤τⁿ

Z τⁿ t∧τⁿ

e^λ%ns(Y_s−ⁿ − a)Z_s−ⁿ dW_sⁿ    

≤ CE

Z T 0

e^2λ%ns|Y_s−ⁿ − a|²kZ_s−ⁿ k²d%ⁿ_s

^1/2

≤ 1 2E sup

t≤τⁿ

e^λt|Y_tⁿ− a|²+ CE Z _T

0

e^λ%ntkZ_t−ⁿ k²d%ⁿ_t. Therefore taking supremum in (7) and using (8) we get estimate on E sup_t≤τne^λt|Y_t∧τⁿ n − a|².

Now, note that from

K_t∧τ^{n n} = Y₀ⁿ− Y_t∧τⁿ n− Z t∧τⁿ

0

f (%ⁿ_s−, Y_s−ⁿ, Z_s−ⁿ )d%ⁿ_s

− Z t∧τⁿ

0

ϕ(%ⁿ_s−, Y_s−ⁿ)dΛ^n,L_s + Z t∧τⁿ

0

Z_s−ⁿ dW_sⁿ

and the previous estimates it follows that E sup_t≤τne^λt|K_tⁿ|² + Ee^λτn|Kⁿ|_τⁿ ≤ CE



e^λτn|ξⁿ− a|²+ (Λ^n,L_τn )²

+ Z τⁿ

0

e^λ%ns|f (%ⁿ_s−, a, 0)|²d%ⁿ_s + Z τⁿ

0

e^λ%ns|ϕ(%ⁿ_s−, a)|²dΛ^n,L_s

 ,

which shows the part (a).

(b) Let (Yⁿ, Zⁿ, Kⁿ), ( ˜Yⁿ, ˜Zⁿ, ˜Kⁿ) be two solutions of (5). By the Itˆo formula,

e^λ(t∧τn)|Y_t∧τⁿ n− ˜Y_t∧τⁿ n|²+ λ Z τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ − ˜Y_s−ⁿ|²d%ⁿ_s + Z τⁿ

t∧τⁿ

e^λ%nskZ_s−ⁿ − ˜Z_s−ⁿ k²d%ⁿ_s

≤ (2µ + L²/ε) Z τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ − ˜Y_s−ⁿ|²d%ⁿ_s + ε Z τⁿ

t∧τⁿ

e^λ%nskZ_s−ⁿ − ˜Z_s−ⁿ k²d%ⁿ_s

+ 2β Z τⁿ

t∧τⁿ

e^λ%ns|Y_s−ⁿ− ˜Y_s−ⁿ|²dΛ^n,L_s − 2 Z τⁿ

t∧τⁿ

e^λ%ns(Y_s−ⁿ − ˜Y_s−ⁿ)(Z_s−ⁿ − ˜Z_s−ⁿ )dW_sⁿ.

Integrating and choosing ε < 1 such that 2µ + L²/ε < λ we complete the proof.

Proof of Theorem 2. The proof of the theorem we divide into three steps.

Step 1. For every natural M we construct a solution (Y^M, Z^M, K^M) of GRBSDE

Y_t^M = ξ + Z M ∧τ

t∧τ

f (s, Y_s^M, Z_s^M)ds + Z M ∧τ

t∧τ

ϕ(s, Y_s^M)dΛs

(9)

− Z τ

t∧τ

Z_s^MdWs+ K_{M ∧τ}^M − K_t∧τ^M , t ∈ R⁺

in the following way. Let ξ_M = E(ξ|F_M) and Λ^τ be a process stopped in the stopping time τ , i.e., Λ^τ_t = Λt∧τ.

For t ∈ [0, M ] consider

Y_t^M = ξ_M + Z M

t

1_{[0,τ ]}(s)f (s, Y_s^M, Z_s^M)ds + Z M

t

ϕ(s, Y_s^M)dΛ^τ_s (10)

− Z _M

t

Z_s^MdW_s+ K_M^M − K_t^M.

Since ξM is FM measurable, from [7] it follows that there exists a unique solution of (10) on a deterministic interval [0, M ]. Note that, on the set {t ≥ τ }, ξM = ξ and

Y_t^M = ξ + 0 − Z M

t

Z_s^MdW_s+ K_M^M − K_t^M.

By the uniqueness of the solution of BSDE, increments of the process K^M are constant, i.e., K_M^M = K_t^M = K_τ^M. Therefore Y_t^M = ξ − RM

t Z_s^MdW_s and in particular Y_τ^M = E(ξ|Fτ) = ξ. On the other hand, by the Itˆo formula

E



|Y_τ^M|²+ Z M

τ

kZ_s^Mk²ds



= E|ξ|².

As a consequence, on the set {t ≥ τ } Y_t^M = ξ, and Z_t^M = 0.

For t > M put Y_t^M = ξ_t, Z_t^M = ζ_t and K_t^M = K_M^M. These processes satisfy Y_t^M = ξ −

Z τ t

Z_s^MdWs,

and on the set {t ≥ τ }, Y_t^M = ξt= ξ and Z_t^M = 0.

It can be shown that (compare [8]) E

 sup

t≤τ

e^λt|Y_t^M − a|²+ Z τ

0

e^λt(|Y_t^M − a|²dΓ_t+ kZ_t^Mk²dt) + Z τ

0

e^λtd|K^M|_t



≤ CE



e^λτ|ξ − a|²+ Z τ

0

e^λt|f (t, a, 0)|²dt + Z τ

0

e^λt|ϕ(t, a)|²dΛt

 , (11)

where Γt= Λt+ t. It is clear that the solution of (9) converges to the solution of (2) when M → ∞.

Step 2. For every M ∈ N we construct a sequence (Y^n,M, Z^n,M, K^n,M) of solutions of GRBSDE on [0, M ] which approximate the solution of (10) (compare [7]). We will show that this sequence converges to the solution of (5), when M → ∞. Let ξ_Mⁿ = E(ξⁿ|F_Mⁿ). For j = nM we put y_τ^n,Mn

j = ξ_Mⁿ , z_τ^n,Mn j = 0,

∆k_τ^n,Mn

j+1 = 0. Next, for j = nM − 1, . . . , 0 we solve y_τ^n,Mn

j = y_τ^n,Mn j+1+ 1

nf

j/n, y_τ^n,Mn j , z_τ^n,Mn

j



1_{[nτⁿ_]>j}

+ ϕ

j/n, y_τ^n,Mn j



∆Λ^n,L_τn

j 1_{[nτⁿ_]>j}− z^n,M_τn j ∆W_τⁿⁿ

j+1 + ∆k_τ^n,Mn j+1.

If we put Y_t^n,M = y_τ^n,Mn

[nt], Z_t^n,M = z_τ^n,Mn

[nt] and K_t^n,M = P

j≤[nt]∆k_τ^n,Mn

j , then the triple (Y^n,M, Z^n,M, K^n,M) is a unique (compare Proposition (3)) solution of

Y_t^n,M = ξⁿ_M+

Z _{M ∧τ}ⁿ

t∧τⁿ

f (%ⁿ_s−, Y_s−^n,M, Z_s−^n,M)d%ⁿ_s +

Z _{M ∧τ}ⁿ

t∧τⁿ

ϕ(%ⁿ_s−, Y_s−^n,M)dΛ^n,L_s

− Z M

t∧τⁿ

Z_s−^n,MdW_sⁿ+ K_{M ∧τ}^n,Mn− K_t∧τ^n,Mn, t ∈ [0, M ].

(12)

Let (Yⁿ, Zⁿ, Kⁿ) be a solution of (5). For t ∈ [0, M ] we have

Y_t∧τⁿ n = Y_{M ∧τ}ⁿ n+

Z M ∧τⁿ t∧τⁿ

f (%ⁿ_s−, Y_s−ⁿ, Z_s−ⁿ )d%ⁿ_s +

Z M ∧τⁿ t∧τⁿ

ϕ(%ⁿ_s−, Y_s−ⁿ)dΛ^n,L_s

−

Z M ∧τⁿ t∧τⁿ

Z_s−ⁿ dW_sⁿ+ K_{M ∧τ}ⁿ n− K_t∧τⁿ n.

By the Itˆo formula and assumptions,

e^λ(t∧τn)|Y_tⁿ− Y_t^n,M|²+

Z M ∧τⁿ t∧τⁿ

e^λ%ns(λ|Y_s−ⁿ − Y_s−^n,M|²+ kZ_s−ⁿ − Z_s−^n,Mk²)d%ⁿ_s

≤ e^{λ(M ∧τn)}|Y_{M ∧τ}ⁿ n− ξ_Mⁿ|²+ (2µ + L²/ε)

Z M ∧τⁿ t∧τⁿ

e^λ%ns|Y_s−ⁿ − Y_s−^n,M|²d%ⁿ_s

+ ε

Z M ∧τⁿ t∧τⁿ

e^λ%nskZ_s−ⁿ − Z_s−^n,Mk²d%ⁿ_s + 2β

Z M ∧τⁿ t∧τⁿ

e^λ%ns|Y_s−ⁿ − Y_s−^n,M|²dΛ^n,L_s (13)

− 2

Z M ∧τⁿ t∧τⁿ

e^λ%ns(Y_s−ⁿ − Y_s−^n,M)(Z_s−ⁿ − Z_s−^n,M)dW_sⁿ.

Choosing ε < 1 such that 2µ + L²/ε < λ and after integrating we get

E



e^λ(t∧τn)|Y_tⁿ− Y_t^n,M|²+

Z M ∧τⁿ t∧τⁿ

e^λ%ns|Y_s−ⁿ − Y_s−^n,M|²dΛ^n,L_s

+

Z M ∧τⁿ t∧τⁿ

e^λ%ns(|Y_s−ⁿ − Y_s−^n,M|²+ kZ_s−ⁿ − Z_s−^n,Mk²)d%ⁿ_s



≤ CEe^{λ(M ∧τn)}|Y_{M ∧τ}ⁿ n− ξ_Mⁿ|².

Moreover, taking supremum in (13) we get E sup

t≤M

e^λ(t∧τn)|Y_tⁿ− Y_t^n,M|²≤ CEe^{λ(M ∧τn)}|Y_{M ∧τ}ⁿ n− ξ_Mⁿ|².

Similarly we can compute for λ⁰ such 2µ + L²/ε = λ⁰< λ. Since sup

n

Ee^{λ(M ∧τn)}|Y_{M ∧τ}ⁿ n− ξ_Mⁿ|² ≤ 2 sup

n

E sup

t≤τⁿ

e^λt|Y_tⁿ|²+ 2 sup

n

Ee^λτn|ξⁿ|² < ∞

and Ee^{λ0(M ∧τn)}|Y_{M ∧τ}ⁿ n− ξⁿ_M|² ≤ Ce^(λ0−λ)MEe^{λ(M ∧τn)}|Y_{M ∧τ}ⁿ n− ξⁿ_M|², we have

M →∞lim sup

n

E

 sup

t≤τⁿ

e^λ0t|Y_tⁿ− Y_t^n,M|²+

Z _{M ∧τ}ⁿ

0

e^λ0%nskZ_s−ⁿ − Z_s−^n,Mk²d%ⁿ_s

+

Z M ∧τⁿ 0

e^λ0%ns|Y_s−ⁿ − Y_s−^n,M|²dΛ^n,L_s + sup

t≤τⁿ

|K_tⁿ− K_t^n,M|²



= 0.

Step 3. Using the proof of Theorem 4.1 in [7] we get

n→∞lim E

 sup

t≤M

|Y_t∧τ^n,Mn− Y_t∧τ^M |²+ Z M

0

kZ_t−^n,M,τn− Z_t^M,τk²dt

+ Z M

0

|Y_t−^n,M,τn− Y_t^M,τ|²dΛt+ sup

t≤M

|K_t∧τ^n,Mn− K_t∧τ^M |²



= 0.

Combining arguments from steps 1–3 we complete the proof.

4. Partial differential equations

In [7] and [16] it was shown that GRBSDE with deterministic terminal time gives a probabilistic formula for the viscosity solution to an obstacle problem for parabolic PDE with Neumann boundary condition. Moreover, in [7] the appli- cation of the discrete approximation in solving appropriate PDE was given. In [8] the connection between GRBSDE with random terminal time and an obstacle problem for elliptic PDE with Neumann boundary condition was shown. Here we will give an application of the numerical scheme in solving PDE.

Let O, G be open connected bounded and smooth subsets of R^m such that G ∩ O 6= ∅ and ∂O ∩ G 6= ∅, ∂G ∩ O 6= ∅.

Assume that b : R^m→ R^m and σ : R^m→ R^m×m are Lipschitz functions, i.e.

for some L⁰ > 0, all x, x⁰ ∈ R^m

(B1) |b(x) − b(x⁰)| + kσ(x) − σ(x⁰)k ≤ L⁰|x − x⁰|

and let (X^x, A^x) be a solution of SDE with reflection, i.e., X_t^x= x +

Z t 0

b(X_s^x)ds + Z t

0

σ(X_s^x)dWs+ A^x_t, t ∈ R⁺, (14)

where P (X^x ∈ ¯O) = 1, A^x is a process with locally finite variation |A^x|, that increases only if X_t^x ∈ ∂O; A^x₀ = 0, X0 = x ∈ O ∩ G (for a unique existence see [9]).

In particular, if O = {x; φ(x) > 0}, ∂O = {x; φ(x) = 0} for some φ ∈ C_b²(R^m) such that |∇φ(x)| = 1 for x ∈ ∂O, then

A^x_t = Z t

0

∇φ(X_s^x)d|A^x|_s = Z t

0

∇φ(X_s^x)1_{X^x

s∈∂O}d|A^x|_s. Define τ^x= inf{t ≥ 0; X_t^x∈ G} and assume that/

(B2) Eτ^x< ∞.

Suppose that D = (a₁, b₁) × (a₂, b₂) × . . . × (a_d, b_d) and g : ∂G ∩ ¯O → ¯D. Let (Y^x, Z^x, K^x) be a solution of GRBSDE with data (τ^x, g(X_τ^xx), F, Φ, |A^x|), where F (t, ω, y, z) = f (X_t^x(ω), y, z), Φ(t, ω, y) = ϕ(X_t^x(ω), y), t ∈ R⁺, ω ∈ Ω, y ∈ R^d, z ∈ R^d×m, i.e.

Y_t∧τ^x x = g(X_τ^xx) + Z τ^x

t∧τ^x

f (X_θ^x, Y_θ^x, Z_θ^x)dθ + Z τ^x

t∧τ^x

ϕ(X_θ^x, Y_θ^x)d|A^x|_θ

− Z τ^x

t∧τ^x

Z_θ^xdWθ+ K_τ^xx − K_t∧τ^x x, t ∈ R⁺. (15)

Assume that functions g, f and ϕ are continuous and there exist constants κ, p ≥ 0, L > 0, µ ∈ R and β < 0 such that µ + L²< 0 and for any x ∈ R^m, y, y⁰ ∈ R^d, z, z⁰ ∈ R^d×m,

|g(x)| ≤ κ(1 + |x|^p), hy − y⁰, f (x, y, z) − f (x, y⁰, z)i ≤ µ|y − y⁰|²,

|f (x, y, z) − f (x, y⁰, z⁰)| ≤ L |y − y⁰| + kz − z⁰k
, (B3) |f (x, y, 0)| ≤ κ(1 + |y|), |ϕ(x, y)| ≤ κ,

|ϕ(x, y) − ϕ(x, y⁰)|i ≤ L|y − y⁰|, hy − y⁰, ϕ(x, y) − ϕ(x, y⁰)i ≤ β|y − y⁰|²,

E Z τ^x

0

|f (X_t^x, ξ_t, ζ_t)|²dt < ∞,