Approximation scheme for solutions of backward stochastic differential equations via the representation theorem

BLAISE PASCAL

Mohamed El Otmani

Approximation scheme for solutions of backward stochastic differential equations via the representation theorem

Volume 13, n^o1 (2006), p. 17-29.

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Approximation scheme for solutions of backward stochastic differential equations via the

representation theorem

Mohamed El Otmani

Abstract

We are interested in the approximation and simulation of solutions for the backward stochastic differential equations. We suggest two approximation schemes, and we study theL²induced error.

1. Introduction

Backward Stochastic Differential Equations (BSDEs in short) have been introduced by Pardoux and Peng [16]. One of the main motivations for studying the BSDE’s has been to use it to solve diverse problems in mathe- matical finance [8], in optimal stochastic control [11] and stochastic games [6].

In this paper, we propose to simulate the solution(X, Y, Z)of the coupled Forward-Backward SDE defined for all 0≤t≤T by

(F BSDE)

( Xt=x+^R₀^tb(s, Xs)ds+^R₀^tσ(s, Xs)dWs F orward Y_t=g(X_T) +^R_t^Tf(s, X_s, Y_s, Z_s)ds−^R_t^TZ_sdW_s Backward where x ∈ R and W is a Brownian motion defined on some complete filtered probability space(Ω,F,P,(F_t)0≤t≤T). The forward componentX will be approximated by the Euler or the Milshtein Scheme. The simulation of the couple (Y, Z) is more delicate. Douglas et al presented in [7] a numerical method for a class of Forward-Backward SDEs based on the finite difference approximation of the associated PDE and the four steps scheme developed in [13]. In [5], Chevance proposed a numerical method for BSDE’s where the generatorf is not dependent on the control gradient Z. The difficulty is in the approximation of the processZ that comes only from the martingale representation Theorem. In the case wheref depends only on Z, Bally has developed in [1] a discretization scheme considers a

random time partition. His method is more difficult to implement. The work of Zhang [17] is more interesting but doesn’t answer the question of approximation of Z. Bouchard and Touzi [3] invested this work to give an implicit approximation scheme. Recently, Gobet et al [10] propose a new numerical scheme based on iterative regression function basis which coefficients are evaluated using Monte Carlo simulation.

Our approach is to give an approximation scheme via the representation for the solution(Y, Z)of the Backward SDE. We suggest a pseudo approx- imation scheme then an explicit discretization scheme, and we calculate the L² error induced to these approximations.

Throughout the paper, we use the following notation:

Notation 1.1. Let π : 0 =t₀ < t₁ < ... < t_n =T be a regular partition of the time interval [0, T]such that |π|=T /n and ti =i|π|for i= 0, ..., n.

We denote by ∆Wti+1 =Wti+1−Wti and Ei = E[./F_ti]for all i. Finally, C will denote a generic constant independent ofπ that may take different values from line to line.

We shall make use of the following assumptions:

Condition 1. σ ∈ C_b^0,2([0, T]×R,R^∗) and b ∈ C_b^0,1([0, T]×R,R), and all derivatives (respect to x) are uniformly bounded by a common constant K > 0. Further, there exists a constant κ > 0 such that for any x ∈ R and t∈[0, T] κ≤ |σ(t, x)| ≤K and |b(t,0)| ≤K.

Condition 2. The functionsf ∈ C([0, T]×R×R×R,R) andg∈ C(R,R) are uniformly K-Lipschitz i.e ∀x, x⁰, y, y⁰, z, z⁰ ∈R andt, t⁰∈[0, T]

|f(t, x, y, z)−f(t⁰, x⁰, y⁰, z⁰)| ≤K |t−t⁰|+|x−x⁰|+|y−y⁰|+|z−z⁰| and

|g(x)−g(x⁰)| ≤K|x−x⁰|.

Under these assumptions, there exists a unique adapted process(X, Y, Z) solution of the (F BSDE)such that

E sup

0≤t≤T

|X_t|²+ sup

0≤t≤T

|Y_t|²+ Z T

0

|Z_s|²ds

!

≤C(1 +|x|²), (1.1) The following estimates are standard [17]

1≤i≤nmax sup

t∈(ti−1,ti]

E

|X_t−X_ti−1|²+|Y_t−Y_ti−1|²≤C(1 +|x|²)|π|, (1.2)

and

n

X

i=1

E Z ti

ti−1

|Z_s−Zti−1|²+|Z_s−Zti|²ds≤C(1 +|x|²)|π|. (1.3) We shall cite main representation of the solution(X, Y, Z).

To begin with, let us define for 0< t < r < T the process N_r^t= 1

r−t Z _r

t

σ⁻¹(s, X_s)∇X_sdW_s(∇X_t)⁻¹, where ∇X is the unique solution of the following Linear SDE

∇X_t= 1 + Z t

0

∂xb(s, Xs)∇X_sds+ Z t

0

∂xσ(s, Xs)∇X_sdWs. It is known that ∇X is invertible and

(∇X_t)⁻¹ = 1− Z t

0

∂_xb(s, X_s)− |∂_xσ(s, X_s)|²(∇X_s)⁻¹ds

− Z t

0

∂_xσ(s, X_s)(∇X_s)⁻¹dW_s.

As a consequence of these notations, one has the following estimates [15]:

E|N_r^t|²

1

2 ≤ C

√r−t, E|N_r^t₁−N_r^t₂|² ≤C |r₁−r₂|

(r1−t)(r2−t). (1.4) The representation formula can be written P- almost surely [14]:







Yt=E

g(X_T) +^R_t^Tf(s, Xs, Ys, Zs)ds/F_t, Zt=E

g(XT)N_T^t +^R_t^T f(s, Xs, Ys, Zs)N_s^tds/F_tσ(t, Xt).

Remark 1.2. A direct consequence of the representation formula of the matringale integrand Z that: If condition 1 and condition 2 hold; for all p≥2, there exists a constantC_p >0 depending only onT, K and psuch that

E sup

0≤t≤T

|Z_t|^p ≤Cp(1 +|x|^p).

(see [15] Lemma 2.5).

Now, we are ready to give the approximation schemes.

2. Backward approximation schemes

2.1. Pseudo-discretization

In this part, we consider that the forward component X will be approxi- mated by the classical Euler scheme

X¯₀=x,

X¯ti+1 = ¯Xti+b(ti,X¯ti)|π|+σ(ti,X¯ti)∆Wti+1 for any i= 0, ..., n−1.

We set for all t_i ≤t≤t_i+1

X¯_t= ¯X_ti+b(t_i,X¯_ti)(t−t_i) +σ(t_i,X¯_ti)(W_t−W_ti), and we have the following estimates for all p≥2

E sup

0≤t≤T

|X¯t|^p<+∞ and E sup

0≤t≤T

|X_t−X¯t|^p≤Cp|π|^p/2. (2.1) To approximate the solution of the Backward SDE, we consider the natural discrete time scheme







Y¯_tn =g( ¯X_tn), and for i=n−1, ...,0

Y¯_ti =EiY¯_ti+1+f(t_i+1,X¯_ti+1,Y¯_ti+1,Z¯_ti+1)|π|

Z¯ti = Ei

hY¯ti+1N_t^t_i+1ⁱ +f(ti+1,X¯ti+1,Y¯ti+1,Z¯ti+1)N_t^t_i+1ⁱ |π|ⁱσ(ti,X¯ti), By a backward induction argument, we can see that ( ¯Y_ti,Z¯_ti) ∈ L² for any i, and they are deterministic functions of X¯_ti. For later use, we need to introduce a continuous time approximation for (Y, Z). Since, from the classical martingale representation Theorem, there exists a predictable process Z˜∈L²([0, T]×Ω)such that

Y¯ti+1+f(ti+1,X¯ti+1,Y¯ti+1,Z¯ti+1)|π|= ¯Yti+ Z ti+1

ti

Z˜sdWs. We then define for all t∈(t_i, t_i+1)

( Y¯t= ¯Yti+1+f(ti+1,X¯ti+1,Y¯ti+1,Z¯ti+1)(ti+1−t)−^R_t^ti+1Z˜sdWs

Z¯t= ¯Zti. (2.2)

The following Theorem provides the error estimates due to this approxi- mation

Theorem 2.1. There exists a constant C >0 depending only on T and K such that

sup

0≤t≤TE|Y_t−Y¯_t|²+E Z T

0

|Z_t−Z¯_t|²dt≤C|π|.

Proof. We denoteΘ_s= (X_s, Y_s, Z_s)and Θ¯_s= ( ¯X_s,Y¯_s,Z¯_s). From the Itô’s formula and the Lipschitz property of f, we get

E|Y_t−Y¯_t|²+E Z ti+1

t

|Z_s−Z˜_s|²ds = E|Y_ti+1−Y¯_ti+1|² + 2E

Z ti+1

t

(Ys−Y¯s)(f(s,Θs)−f(ti+1,Θ¯ti+1))ds

≤ E|Y_ti+1−Y¯_ti+1|² +C

αE Z ti+1

t

|s−t_i+1|²+|X_s−X_ti+1|²+|X_ti+1−X¯_ti+1|²ds

+αE Z ti+1

t

|Y_s−Y¯_s|²ds+C αE

Z ti+1

t

|Y_s−Y_ti+1|²+|Y_ti+1−Y¯_ti+1|²ds

+C αE

Z ti+1

t

|Z_s−Z_ti+1|²+|Z_ti+1−Z¯_ti+1|²ds

for every t ∈ (ti, ti+1) and α > 0. Applying the Gronwall Lemma, we obtain by (1.2)

E|Y_t−Y¯t|²≤

(1 +C|π|)E|Y_ti+1−Y¯ti+1|²+C αΛi

e^α|π| (2.3) where Λ_i =|π|²+E^R_t^t_iⁱ⁺¹|Z_s−Z_ti+1|²ds+|π|E|Z_ti+1−Z¯_ti+1|².

In particular for t=ti, we have E|Y_ti−Y¯ti|²+E

Z ti+1

ti

|Z_s−Z˜s|²ds

≤(1 +Cα|π|)

(1 +C|π|)E|Y_ti+1−Y¯_ti+1|²+C αΛ_i

.

Iterating the last inequality to get

E|Y_ti−Y¯_ti|² + E Z tn

ti

|Z_s−Z˜_s|²ds

≤ (1 +C|π|)ⁿ⁻ⁱ(1 +Cα|π|)ⁿ⁻ⁱE|g(X_tn)−g( ¯X_tn)|² + (1 +Cα|π|)C

α

n−i−1

X

j=0

(1 +C|π|)^j(1 +Cα|π|)^jΛ_i+j,

≤ C



E|X_tn−X¯tn|²+|π|+

n−1

X

j=0

E Z tj+1

tj

|Z_s−Ztj+1|²ds





+ (1 +Cα|π|)C α|π|

n−1

X

j=0

E|Z_tj−Z¯_tj|²,

≤ C|π|+ (1 +Cα|π|)C α|π|

n−1

X

j=0

E|Z_tj−Z¯tj|² (2.4)

by the condition 2, (1.3) and (2.1). On the other hand, from the expression of Z and Z, if we denote by¯

Σj =|σ⁻¹(tj, Xtj)Ztj −σ⁻¹(tj,X¯tj) ¯Ztj| we can write

Σ_j =Ej

h(Y_tj+1−Y¯_tj+1)N_t^tj

j+1

+ Z tj+1

tj

f(s,Θs)Ns^tj−f(tj+1,Θ¯tj+1)N_t^t_j+1^j dsⁱ

=Ej

"

Y_tj−Y¯_tj + Z tj+1

tj

(Z_s−Z˜_s)dW_s

# N_t^tj

j+1

+Ej

Z tj+1

tj

f(s,Θ_s)(Ns^tj−N_t^tj

j+1)ds.

Since (2.2) and (1.4) we have Σ_j ≤ E

Z tj+1

tj

|Z_s−Z˜_s|²ds

!¹₂

E|N_t^tj

j+1|²

1 2

+

Z _tj+1

tj

Ej|f(s,Θ_s)|²

1 2

E|N_s^tj−N_t^t_j+1^j |²

1 2 ds,

≤ C

p|π| E Z tj+1

tj

|Z_s−Z˜_s|²ds

!¹₂

+ C Ej(1 + sup

0≤s≤T

|Θ_s|²)

!¹₂ Z tj+1

tj

s t_j+1−s

|π|(s−t_j)ds, (2.5)

≤ C

p|π| E Z tj+1

tj

|Z_s−Z˜_s|²ds

!¹₂

+C Ej(1 + sup

0≤s≤T

|Θ_s|²)

!¹₂ q|π|.

The Lipschitz property of σ and the condition 1 insure that

|Z_tj −Z¯_tj| ≤C|Z_tj||X_tj −X¯_tj|+|Σ_j|. Taking the L² estimates and replacing by (2.5), then we have

|π|

n−1

X

j=0

E|Z_tj−Z¯_tj|² ≤C

n−1

X

j=0

(E|X_tj−X¯_tj|⁴)¹²

+E Z tj+1

tj

|Z_s−Z˜s|²ds+|π|²

≤CE Z T

0

|Z_s−Z˜_s|²ds+C|π|.

(2.6) Now, we plug (2.6) in (2.4) to obtain for i= 0

|Y₀−Y¯₀|²+E Z T

0

|Z_s−Z˜_s|²ds≤C|π|+ (1 +Cα|π|) C

|π|E Z T

0

|Z_s−Z˜_s|²ds.

For α sufficiently larger to C we have ^Pn−1_j=0 E|Z_tj −Z¯_tj|² ≤ C, and we conclude by (2.4) and (2.3) that

sup

0≤t≤TE|Y_t−Y¯_t|² ≤C|π|.

Together, with (1.3) we conclude that E

Z T 0

|Z_s−Z¯s|²ds ≤ 2

n−1

X

j=0

E Z tj+1

tj

|Z_s−Ztj|²ds+ 2|π|

n−1

X

j=0

E|Z_tj−Z¯tj|²,

≤ C|π|.

This completes the proof of the Theorem.

Remark 2.2. We can obtain a similar result if we replace in the approx- imation of the forward component X the Euler scheme by the Milshtein scheme.

In this scheme, the process N is only dependent on the forward com- ponent X, we henceforth call it a pseudo-approximation. The following backward scheme raises this handicap.

2.2. Explicit Approximation scheme

In this section we consider that the processus X will be approximated by the Milshtein scheme described by











X₀^π = x,

X_t^π_i+1 = X_t^π_i + (b−¹₂∂xσσ)(ti, X_t^π_i)|π|+σ(ti, X_t^π_i)∆Wti+1

+ 1

2(∂_xσσ)(t_i, X_t^π_i)(∆W_ti+1)². For t∈(t_i, t_i+1), let us put

X_t^π = X_t^π_i + (b−1

2∂_xσσ)(t_i, X_t^π_i)(t−t_i) +σ(t_i, X_t^π_i)(W_t−W_ti) + 1

2(∂_xσσ)(t_i, X_t^π_i)(W_t−W_ti)². It is known that [9]

∀ p≥2, sup

0≤t≤TE|X_t−X_t^π|^p≤K(T)|π|^p (2.7) where K is an increasing function. Let us define the processes ∇X^π and (∇X^π)⁻¹ by∇X_t^π

0 = (∇X_t^π

0)⁻¹ = 1, and for everyt∈(ti, ti+1]

∇X_t^π = ∇X_t^π

i + (∂xb−1

2∂xσ²)(ti, X_t^π_i)∇X_t^π

i(t−ti) + ∂xσ(ti, X_t^π_i)∇X_t^π

i(Wt−Wti) +|∂_xσ(ti, X_t^π_i)|²∇X_t^π

i(Wt−Wti)²

and

(∇X_t^π)⁻¹ = (∇X_t^π

i)⁻¹−(∂xb−1

2∂xσ²)(ti, X_t^π_i)(∇X_t^π

i)⁻¹(t−ti)

− ∂xσ(ti, X_t^π_i)(∇X_t^π

i)⁻¹(Wt−Wti) + 1

2|∂_xσ(ti, X_t^π_i)|²(∇X_t^π

i)⁻¹(Wt−Wti)².

Now, we are ready to approximate the processN. We put fori= 0, ..., n−1 N_t^π,t_i+1ⁱ = 1

|π|

Z _ti+1

ti

σ⁻¹(s, X_s^π)∇X_s^πdW_s(∇X_t^π_i)⁻¹, and we have the following estimates

Proposition 2.3.

• sup_0≤t≤T E |∇X_t− ∇X_t^π|²+|(∇X_t)⁻¹−(∇X_t^π)⁻¹|²≤C|π|²,

• max0≤i≤n−1E|N_t^ti

i+1−N_t^π,t_i+1ⁱ|² ≤C|π|.

Proof. We denote ∇X¯ the Milshtein approximation of ∇X defined by

∇X¯₀= 1, and

∇X¯_t = ∇X¯_ti+ (∂_xb−1

2∂_xσ)(t_i, X_ti)∇X¯_ti(t−t_i)

+ ∂_xσ(t_i, X_ti)∇X¯_ti(W_t−W_ti) +|∂_xσ(t_i, X_ti)|²∇X¯_ti(W_t−W_ti)² for t∈(ti, ti+1). Then we havesup_0≤t≤TE|∇X_t− ∇X¯t|² ≤C|π|². On the other hand

E|∇X¯_t− ∇X_t^π|² ≤ C(1 +|π|+|π|²)E|∇X¯_ti− ∇X_t^π

i|²

+ C|π|(1 +C|π|)(E|∇X¯_ti|⁴)¹²E|X_ti−X_t^π_i|⁴

1 2

+ C|π|²(E|∇X¯_ti|⁴)1

2|EX_ti−X_t^π_i|⁸

1 2.

In particular for t=t_i+1 E|∇X¯ti+1− ∇X_t^π

i+1|² ≤ (1 +C|π|)E|∇X¯ti− ∇X_t^π

i|²+C|π|³,

≤ C|π|².

The result is followed by the classical iteration.

By the same argument, we prove that E|(∇X_t)⁻¹−(∇X_t^π)⁻¹|²≤C|π|². For the second point, we have

|π|²E|N_t^ti

i+1−N_t^π,ti

i+1|² ≤CⁿE|(∇X_ti)⁻¹−(∇X_t^π_i)⁻¹|^2R_t^ti+1

i E|∇X_s|²ds +E|(∇X_t^π

i)⁻¹|² Z ti+1

ti

E|∇X_s−∇X_s^π|²ds

+E|(∇X_t^π

i)⁻¹|² Z ti+1

ti

E|X_s−X_s^π|⁴

1 2

E|∇X_s|²

1 2 ds

,

≤C|π|³

by the condition 1 and the first point. This completes the proof of the

proposition.

Now, we present our explicit backward approximation scheme. We con- sider the process (Y^π, Z^π) defined by











Y_t^π_n =g(X_t^π_n), and for i=n−1, ...,0 Y_t^π_i =Ei

hY_t^π_i+1+f(t_i+1, X_t^π_i+1, Y_t^π_i+1, Z_t^π_i+1)|π|ⁱ Z_t^π_i = Ei

hY_t^π_i+1N_t^π,t_i+1ⁱ +f(t_i+1, X_t^π_i+1, Y_t^π_i+1, Z_t^π_i+1)N_t^π,t_i+1ⁱ|π|ⁱσ(t_i, X_t^π_i).

The continuous version is defined in the same way that the pseudo-approxi- mation:

Y_t^π =Y_t^π_i+1+f(ti+1, X_t^π_i+1, Y_t^π_i+1, Z_t^π_i+1)(ti+1−t)−

Z ti+1

t

Z˜_s^πdWs, Z_t^π =Z_t^π_i where Z˜^π is the unique square integrable predictable process verifying

Y_t^π_i+1 =Ei[Y_t^π_i+1] + Z ti+1

ti

Z˜_s^πdW_s.

With these suggestions, we provide the induce error of this approximation.

Theorem 2.4.

sup

0≤t≤TE|Y_t−Y_t^π|²+E Z T

0

|Z_t−Z_t^π|²dt≤C|π|

where C >0 is only dependent on K, κ and T. Proof. To prove the Theorem, it suffices to show that

sup

0≤t≤TE|Y¯_t−Y_t^π|²+E Z T

0

|Z¯_s−Z_s^π|²ds≤C|π|.

We denote Θ^π_s = (X_s^π, Y_s^π, Z_s^π), and let β >0 be a constant to be chosen later on. From the Lipschitz property of f we get

E|Y¯_t−Y_t^π|²+E Z ti+1

t

|Z˜_s−Z˜_s^π|²ds

=E|Y¯_ti+1−Y_t^π_i+1|²+ 2E Z ti+1

t

( ¯Y_s−Y_s^π)f(t_i+1,Θ¯_ti+1)−f(t_i+1,Θ^π_ti+1)ds,

≤βE Z ti+1

t

|Y¯_s−Y_s^π|²ds+C

β|π|E|Z¯_ti+1−Z_t^π_i+1|²+(1+C

β|π|)E|Y¯_ti+1−Y_t^π_i+1|². Using Gronwall Lemma and the classical iteration to obtain

E|Y¯ti−Y_t^π_i|²+E Z tn

ti

|Z˜s−Z˜_s^π|²ds≤(1 +Cβ|π|)C β|π|

n−1

X

j=0

E|Z¯tj−Z_t^π_j|² On the other hand, σ is bounded byK, then we can write

|Z¯tj −Z_t^π_j| ≤KEj

hY¯tj+1N_t^t_j+1^j −Y_t^π_j+1N_t^π,t_j+1^j +|π|f(tj+1,Θ¯tj+1)N_t^t_j+1^j

−f(t_j+1,Θ^π_tj+1)N_t^π,t_j+1^ji

≤KEj

h( ¯Ytj+1−Y_t^π_j+1+|π|f(tj+1,Θ¯tj+1)

−|π|f(t_j+1,Θ^π_tj+1))N_t^t_j+1^{j i} +KEj

h(Y_t^π_j+1+|π|f(t_j+1,Θ^π_tj+1))(N_t^tj

j+1−N_t^π,tj

j+1)ⁱ

≤KE

Z tj+1

tj

( ˜Z_s−Z˜_s^π)dW_s.N_t^t_j+1^j +KEj[Y_t^π_j+1+|π|f(t_j+1,Θ^π_tj+1)]²

1

2(E|N_t^tj

j+1−N_t^π,t_j+1^j|²)¹²

≤ C p|π| E

Z tj+1

tj

|Z˜s−Z˜_s^π|²ds

!¹₂

+C^p|π|Ej(Y_t^π_j+1+|π|f(tj+1,Θ^π_tj+1))²

1 2 .

By the same way that the pseudo-approximation, we prove forβ larger to C that^Pn−1_j=0E|Z¯_tj−Z_t^π_j|² ≤C, and we conclude that

E|Y¯t−Y_t^π|²+E Z T

0

|Z¯s−Z_s^π|²ds ≤ |π|

n−1

X

j=0

E|Z¯tj−Z_t^π_j|²

≤ C|π|.

This completes the proof of the Theorem.

Remark 2.5. The conditional expectation in the above discretization scheme, reduced to the regression on the random variable X¯_ti in the pseudo dis- cretization andX_t^π_i in the explicit scheme. Different methods of regression are developed, see for example [2], [4] and [12].

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Mohamed El Otmani Faculty of Sciences Semlalia Department of Mathematics Cadi Ayyad University BP 2390

Marrakesh MOROCCO

m.elotmani@ucam.ac.ma