Convergence of scheme for decoupled forward

Communications on Stochastic Analysis Volume 9 | Number 1

Article 2

3-1-2015

Convergence of scheme for decoupled forward backward stochastic differential equation Hani Abidi Habib Ouerdiane

Follow this and additional works at: https://digitalcommons.lsu.edu/cosa Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Abidi, Hani and Ouerdiane, Habib (2015) "Convergence of scheme for decoupled forward backward stochastic differential equation," Communications on Stochastic Analysis: Vol. 9 : No. 1 , Article 2. DOI: 10.31390/cosa.9.1.02 Available at: https://digitalcommons.lsu.edu/cosa/vol9/iss1/2

Serials Publications

Communications on Stochastic Analysis Vol. 9, No. 1 (2015) 19-41

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CONVERGENCE OF SCHEME FOR DECOUPLED FORWARD BACKWARD STOCHASTIC DIFFERENTIAL EQUATION HANI ABIDI AND HABIB OUERDIANE

Abstract. We study the convergence rate of Bouchard-Touzi-Zhang scheme (in short B-T-Z) for Decoupled Forward Backward Stochastic Differential Equation, this convergence is controlled by the stability of truncation error and the Markovian property of its processes. Then, we present the algorithm used and provide some numerical results. Finally, we give a fundamental stability property for the reflected Backward Stochastic Differential Equation in a Markovian framework.

1. Introduction We are interested by the discrete time approximation of a decoupled forward backward stochastic differential equation (FBSDE), which has a solution given by the triplet (X,Y,Z) satisfying: ∫ t ∫ t Xt = X0 + b(s, Xs )ds + σ(s, Xs )dWs 0

Yt

0

∫

T

∫

T

f (s, Xs , Ys , Zs )ds −

= g(XT ) + t

Zs dWs

(1.1)

t

where b : [0, T ] × R → R, σ : [0, T ] × R → R, and f : [0, T ] × R × R × R → R are Lipshitz-continuous functions, g : R → R is differentiable with continuous and bounded first derivative, T is a given positive constant and W is a Brownian motion. The BSDE’s study was performed by Pardoux and Peng [10] who proved the existence and the uniqueness of BSDE solution under the Lipshitz continuity assumption. Many BSDE don’t have an explicit solution, however it’s approximated by many schemes, such as the four steps scheme resolved by Ma Protter and al. [6]. More recently, other authors developed some schemes to discretize the BSDE like [1] [7]. Then, El Karoui and al. [5] introduced the reflected BSDE (RBSDE in short) with one continuous lower barrier L = (Lt ). More precisely, a solution of such equation, associated with a terminal value ξ, is a triplet (Yt , Zt , At )0≤t≤T of

Received 2014-5-9; Communicated by the editors. 2010 Mathematics Subject Classification. 60H10, 60H30, 35K85. Key words and phrases. BSDE, reflected BSDE, B-T-Z scheme, Itˆ o Taylor expansion. 19

20

HANI ABIDI AND HABIB OUERDIANE

adapted processes valued in R × R × R+ , satisfying: ∫

∫

T

T

f (Ys , Zs )ds + AT − At −

Yt = ξ + t

Zs dWs , 0 ≤ t ≤ T a.s

(1.2)

t

and Yt ≥ Lt a.s. For any 0 ≤ t ≤ T , At is non-decreasing continuous process. The role of At is to push upward the process Y in a minimal way, in order to keep it ∫T above L = (Lt ). In this way it satisfies 0 (Ys −Ls )dAs = 0. The authors in [5] have proved that the equation (1.2) has a unique solution when ξ is square integrable, f is uniformly Lipschitz with respect to (y, z) and L = (Lt ) is a continuous process. The aim of our study is to reformulate the convergence rate of B-T-Z scheme using another method namely the tree method combining weak Taylor scheme. Chassagneux and D. Crisan [3] performed this kind of method for autonomous functions used in (1.1), i.e, f(t,x,y,z)=f(y,z), b(s,x)=b(x) and σ(s, x) = σ(x). The advantage of this method is: firstly, it does not allow Monte Carlo error because the approximation of the conditional expectation is based on the forward approximation of X by the tree method. Secondly, not like Malliavin method [1] and the regression method [7], this require to estimate only a small number of conditional expectations at each step. Finally, it is more simple to implement. The outline of our paper is presented below. Section 2 is devoted to define the Itˆ o Taylor expansion announced by Kloeden and Platen [8]. In section 3, we study the convergence of B-T-Z scheme for FBSDE which is based on a fundamental stability property, an Itb o Taylor expansion and the approximation of Brownian motion with backward random walk in a discrete time-grid 0 = t0 < t1 < ... < tn = T, n ∈ N. In Section 4, we give some estimates to prove the stability property of reflected backward stochastic differential equation by using the Penalization approximation.

2. Preliminaries and Notations In the sequel, let (Ω, F, P) be a complete probability space, (Wt )t≥0 is a 1dimensional Brownian motion defined on a fixed interval [0, T], with a fixed T > 0. 2.0.1. Notations. For using later, We denote by: • • • •

{F}0≤t≤T the natural filtration generated by the Brownian motion W. Cb : set of bounded process. L2 (Ft ) the space of Ft -meas. random variable ξ such that E[|ξ|2 ] < ∞. S 2 the space of prog. meas process Y such that E[sup0≤t≤T |Yt |2 ] < ∞. ( ∫ ) 21 T • H2 the space of prog. meas process Z such that E 0 |Zs |2 ds < ∞. Now, our effort concentrates on the following FBSDE: ∫t ∫t { Xt = X0 + 0 b(s, Xs )ds + 0 σ(s, Xs )dWs Yt = g(XT ) +

∫T t

f (s, Xs , Ys , Zs )ds −

∫T t

(2.1) Zs dWs .

BSDE AND RBSDE

We relate our FBSDE (2.1) to the parabolic differential equation { L0 u(t, x) + f (t, x, u(t, x), L1 (u)(t, x)) = 0 u(T, x) = g(x)

21

(2.2)

where u : R+ × R → R and L0 =

∂ 1 ∂2 ∂ ∂ +b + σ 2 2 ; L1 = σ . ∂t ∂x 2 ∂ x ∂x

(2.3)

1,2 Theorem 2.1. If u ∈ C { ([0, T ] × R)}solves (2.2), then u(t, Xt ) = Yt and Zt = ∂ σ(t, Xt ) ∂x (t, Xt ) where (Yt , Zt )0≤t≤T is the unique solution of the BSDE (2.1).

Proof. By applying Itˆ o formula to u(t, Xt ), we have: du(t, Xt ) = L0 u(s, Xs )ds + L1 u(s, Xs )dWs

(2.4)

Since u solves (2.2), it follows that −du(t, Xt ) = f (t, Xt , u(t, Xt ), σ(t, Xt )∂x u(t, Xt ))dt − σ(t, Xt )∂x u(t, Xt )dWs . with u(T, XT ) = g(XT ). Thus {u(t, Xt ), σ(t, Xt )∂x u(t, Xt ), t ∈ [0, T ]} is equal to the unique solution of BSDE (2.1), and the result is obtained. □ 2.0.2. Approximation of backward stochastic differential equation. [1] Let X be the solution on [0,T] of the stochastic differential equation: ∫ t ∫ t Xt = X0 + b(s, Xs )ds + σ(s, Xs )dWs (2.5) 0

0

where b : R × R −→ R and σ : R × R −→ R are assumed to be C-Lipschitz i.e |b(t, x) − b(t, y)| + |σ(t, x) − σ(t, y)| < C|x − y|. In order to approximate the above FBSDE (2.1), we introduce first the approximation of Xt . Let π : {0 = t0 < t1 < ... < tn = T } be a partition of the interval [0,T]. Throughout this paper, we shall use the grid notations: hi = |ti − ti−1 |, |π| = max hi , and ∆Wi = Wti − Wti−1 . 0≤i≤n

The forward component X will be approximated by the classical Euler scheme : bt X 0 bt X i

= Xt0 bt = X

i−1

bt )hi + σ(ti−1 , X bt )∆Wi . + b(ti−1 , X i−1 i−1

(2.6)

for i=1,...,n. Under the Lipschitz conditions on b and σ, the following L2 estimate for the error due to the Euler scheme (2.6): [ ] 12 1 bt |2 + max lim sup |π|− 2 E sup |Xt − X sup |Xt − Xti−1 |2 < ∞, (2.7) |π|→0

0≤t≤T

1≤i≤n ti−1 ≤t≤ti

see [8]. We shall denote by {Fti }0≤i≤n is the associated discrete time filtration: Fti = σ(Wtj , j ≤ i)

22

HANI ABIDI AND HABIB OUERDIANE

and E[.|Fti ] = Eti [.] the related conditional expectation. Now we define the discrete time approximation of FBSDE (2.1). We need to introduce a continuoustime approximation of (Y,Z). ∫ ti+1 ∫ ti+1 Yti = Yti+1 + f (s, Xs , Ys , Zs )ds + Zs dWs ti

ti

∫

ti+1

= Eti [Yti+1 +

f (s, Xs , Ys , Zs )ds] ti

Then we define the explicit scheme Y ti = Eti [Yti+1 + hi f (ti+1 , Xti+1 , Yti+1 , Zti+1 )]. To obtain an approximation to Z, we observe that ∫ ti+1 ∫ ti+1 Zs dWs = Yti+1 − Yti + f (s, Xs , Ys , Zs )ds ti

ti

So we define Z ti = Eti [H i Yti+1 ] where H i = h1i (Wti+1 − Wti ). Consider now the following natural explicit discrete-time approximation (Yi , Zi ) of (Yti , Zti ), for i:1...n, [1]: (Yn , Zn )

=

(u(tn , Xtn ), L1 u(tn , Xtn ))

(2.8)

Yi

= Eti [Yi+1 + hi f (ti , Xi+1 , Yi+1 , Zi+1 )]

Zi

= Eti [H i Yi+1 ].

2.0.3. Truncation error. The global errors that we need to control here are ε(Y, π) = sup E[|Yti − Yi | ] and ε(Z, π) = 2

0≤i≤n

n ∑

hi E[|Zti − Zi |2 ]

i=0

To control these errors we will use the local truncation error for the pair (Y, Z) defined as ηi = ηiY + ηiZ where ηiY =

1 |Yt − Y¯ti |2 and ηiZ = |Zti − Z¯ti |2 h2i i

and Y¯ti Z¯t i

= Eti [Yti+1 + hi f (ti , Xti+1 , Yti+1 , Zti+1 )] i

= Eti [H Yti+1 ].

The global truncation error for a given grid π is given by: Γ(π) = ΓY (π) + ΓZ (π) ΓY (π) =

n−1 ∑ i=0

hi ηiY

and ΓZ (π) =

n−1 ∑ i=0

hi ηiZ .

(2.9)

BSDE AND RBSDE

23

3. Forward-Backward Stochastic Differential Equation 3.1. Itˆ o Taylor expansion. This section discusses the stochastic version of Taylor expansion to understand how the stochastic integration methods are obtained. In the next part, we will use some assumptions like: (Hr ) : the coefficient b, σ in (1.1) belongs to Cb3,6 and f belongs to Cb3,6,6,6 and the value function u belongs to Cb3,6 . First, we recall how we can obtain the stochastic version of the Taylor expansion: Let Xt be the solution of this EDS: { dXt = b(t, Xt )dt + σ(t, Xt )dWt (3.1) X(t0 ) = X0 which has the following integral form: ∫ t ∫ t Xt = X0 + b(s, Xs )ds + σ(s, Xs )dWs ∀t ∈ [t0 , T ] t0

(3.2)

t0

Lemma 3.1. For u : [0, T ] × R → R ∈ C 3,6 (R), we have: u(t, Xt )

=

u(t0 , Xt0 ) + L0 u(t0 , Xt0 )(t − t0 ) + L1 u(t0 , Xt0 )(Wt − Wt0 ) ∫ t∫ s ∫ t∫ s +L0 L1 u(t0 , Xt0 ) dzdWs + L1 L1 u(t0 , Xt0 ) dWz dWs ∫ t∫

t0

t0 s

t0

t0

+L0 L0 u(t0 , Xt0 ) where

∫ t∫ R

s

∫

t0

∫ t∫ dzds + L1 L0 u(t0 , Xt0 )

dWz ds + R t0

z

L0 L0 L0 u(z, Xz )drdzds

= t0

∫

t0 t0 t∫ s∫ z

L0 L0 L0 u(z, Xz )dWr dzds

+ t0 ∫ t

t0 ∫ s

t0 ∫ z

L0 L1 L0 u(z, Xz )drdWz ds

+ t0

t0 s

t0 ∫ t

t0 ∫ s

t0 ∫ z

t0 ∫ t

t0 ∫ s

t0 ∫ z

∫ t∫

∫

t0 z

L1 L1 L0 u(z, Xz )dWr dWz ds

+

L0 L0 L1 u(z, Xz )drdzdWs

+

L1 L0 L1 u(z, Xz )dWr dzdWs

+ t0

t0 s

t0 ∫ t

t0 ∫ s

t0 ∫ z

t0

t0

t0

∫ t∫

t0 s

∫

t0 z

L0 L1 L1 u(z, Xz )drdWz dWs

+

L1 L1 L1 u(z, Xz )dWr dWz dWs .

+

t0

24

HANI ABIDI AND HABIB OUERDIANE

Proof. Using the Itˆo formula, we have for u : [t0 , T ] × R → R ∈ C 3,6 (R) and t0 ≤ t ≤ T : ∫ t( ∂ ∂ u(t, Xt ) = u(t0 , Xt0 ) + u(s, Xs ) + b(s, Xs ) u(s, Xs ) ∂t ∂x t0 ) 1 2 ∂2 + σ (s, Xs ) 2 u(s, Xs ) ds 2 ∂x ∫ t ∂ + σ(s, Xs ) u(s, Xs )dWs ∂x t0 ∫ t ∫ t 0 = u(t0 , Xt0 ) + L (u(s, Xs ))ds + L1 (u(s, Xs ))dWs . t0

t0

Applying again the Itˆo formula to L0 u and L1 u, we obtain: ∫ t( u(t, Xt ) = u(t0 , Xt0 ) + L0 u(t0 , Xt0 ) t0 ∫ s ∫ s ) 0 0 + L L (u(z, Xz ))dz + L1 L0 (u(z, Xz ))dWz ds t0

∫ t( + t ∫ 0s

+

t0

∫

s

L1 u(t0 , Xt0 ) +

L0 L1 (u(z, Xz ))dz t0

) L1 L1 (u(z, Xz ))dWz dWs

t0

= u(t0 , Xt0 ) + L0 u(t0 , Xt0 )(t − t0 ) + L1 u(t0 , Xt0 )(Wt − Wt0 ) ∫ t∫ s ∫ t∫ s +L0 L1 u(t0 , Xt0 ) dzdWs + L1 L1 u(t0 , Xt0 ) dWz dWs t0

∫ t∫

t0 s

+L0 L0 u(t0 , Xt0 )

t0

∫ t∫ dzds + L1 L0 u(t0 , Xt0 )

t0

t0

dWz ds + R t0

where ∫ t∫ R

∫

s

z

L0 L0 L0 u(z, Xz )drdzds

= t0

t0

∫ t∫

t0 s

t0 s∫ z

L1 L0 L0 u(z, Xz )dWr dzds

+ t0 ∫ t

t0 ∫ s

t0 ∫ z

t0 ∫ t

t0 ∫ s

t0 ∫ z

L0 L1 L0 u(z, Xz )drdWz ds

+

L1 L1 L0 u(z, Xz )dWr dWz ds

+ ∫ t∫

t0

t0 s

∫

t0 z

t0 ∫ t

t0 ∫ s

t0 ∫ z

t0

t0

t0

L0 L0 L1 u(z, Xz )drdzdWs

+

L1 L0 L1 u(z, Xz )dWr dzdWs

+

t0

BSDE AND RBSDE

∫ t∫

s

∫

25

z

L0 L1 L1 u(z, Xz )drdWz dWs

+ t0

t0 s

t0

t0

∫ t∫

t0 z

∫

L1 L1 L1 u(z, Xz )dWr dWz dWs .

+ t0

□ Remark 3.2. The notation R = O(hp ), for p ≥ 1 means that |R| ≤ λt hp where λt is a positive random variable satisfying E|λpt | ≤ Cp , for all p > 0. 3.2. General convergence results. The aim of this paper is to compute the global numerical error which is the sum of the discrete time approximation error controlled by the rest integral of Itˆ o Taylor expansion given by Theorem 3.7 and the numerical error induced by the approximation of the conditional expectations given in Theorem 3.9. The numerical scheme presented above are still theoretical. We first establish a stability property the scheme. 3.2.1. Stability of the scheme. For this rate of convergence, we are going to introduce the truncation error between the B-T-Z scheme and the perturbed scheme given by: Yei Zei

ei , Yei+1 , Zei+1 )] + ξiY = Eti [Yei+1 + hi f (ti , X = Et [H i Yei+1 ] + ξiZ . i

(3.3) (3.4)

where ξiY and ξiZ are the two perturbed errors belongs in L2 (Fti ) for all i ≤ n. Definition 3.3. (L2 − stability) The scheme given in (3.3) is said to be L2 -stable if max E[|δYi |2 ] + i

n−1 ∑

( [ ] hi E[|δZi |2 ] ≤ C E |π|2 + |δYn |2 + |π||δZn |2

i=0

+

n−1 ∑ i=0

hi E[

) 1 Y 2 Z 2 |ξ | + |ξ | ] , i i h2i

bi , δYi = Yi − Yei and δZi = Zi − Zei , for any sequences ξ Y , where δXi = Xi − X i Z 2 ξi of L (Fti )-random variable and terminal values (Yn , Zn ), (Yen , Zen ) belonging to S 2 × H2 . Theorem 3.4. If f is a Lipshitz-continuous, then the scheme is L2 -stable in the meaning of the definition 3.3, for |π| small enough. Proof. In the following, C > 0 will denote a generic constant may take different values from line to line. Recalling (2.8), by the Cauchy-Schwartz inequality, we

26

HANI ABIDI AND HABIB OUERDIANE

compute, for 1 ≥ η ≥ 0 to be fixed later on that: |δYi |2

|δZi |2

hi η )(Eti |δYi+1 |)2 + Ch2i (1 + )(Eti (|δXi |2 + |δYi+1 |2 (3.5) η hi η Y 2 2 +|δZi+1 | ) + C |ξi | . hi [1 ] (Eti (|δYi+1 |2 ) − (Eti |δYi+1 |)2 ) + |ξiZ |2 . ≤ C (3.6) hi

≤ (1 +

For ϵ > 0, small enough, we get |δYi |2 + ϵhi |δZi |2

hi ≤ (1 + − Cϵ)(Eti |δYi+1 |)2 η [ ] η + Ch2i (1 + ) + Cϵ Eti (|δYi+1 |2 ) hi η η 2 +Chi (1 + )Eti (|δXi |2 ) + C |ξiY |2 + Chi |ξiZ |2 hi hi η 2 2 +Chi (1 + )Eti (|δZi+1 | ). hi

we may use Jensen’s inequality to (Eti |δYi+1 |)2 , we obtain: |δYi |2 + ϵhi |δZi |2

hi η (3.7) + Ch2i (1 + ))(Eti |δYi+1 |2 ) η hi η +Ch2i (1 + )Eti (|δZi+1 |2 ) hi η η +Ch2i (1 + )Eti (|δXi |2 ) + C |ξiY |2 + Chi |ξiZ |2 . hi hi

≤ (1 +

By (2.7) and applying the expectation, E|δYi |2 + ϵhi E|δZi |2

≤

η hi + Ch2i (1 + ))(E|δYi+1 |2 ) η hi η +Ch2i (1 + )E(|δZi+1 |2 ) hi [η ] η +Ch3i (1 + ) + Chi E 2 |ξiY |2 + |ξiZ |2 . hi hi

(1 +

For hi small enough, we can compute that: E|δYi |2 + ϵhi E|δZi |2

≤

[ ] (1 + Chi ) E|δYi+1 |2 + ϵhi E(|δZi+1 |2 ) [η ] η +Ch3i (1 + ) + Chi E 2 |ξiY |2 + |ξiZ |2 . hi hi

Using the discrete version of Gronwall’s Lemma, we obtain sup E|δYi |2 1≤i≤n

n ( ∑ (1 )) ≤ C |π|2 + E[|δYn |2 + |π||δZn |2 ] + hi E 2 |ξiY |2 + |ξiZ |2 ] hi i=1

for small π. This concludes the proof for the Y -part. For the Z-part, the proof is concluded by summing over i in (3.7). □

BSDE AND RBSDE

27

Lemma 3.5. We assume that Hr holds and L0 L0 u, L1 L0 u, L0 L1 u have polynomial growth with respect to X, we get: ∫ ∫ ( ti+1 s 0 0 ) 3 i i)Eti [H L L u(z, Xz )dzds ] = O(|π| 2 ). ii)Eti [H

( i

iii)Eti [H i iv)Eti [H

i

(

ti ti+1

∫

ti

∫

s

ti ∫ ( ti+1

ti ∫ s

ti

ti

∫

ti+1

∫

ti

s

) L0 L1 u(z, Xz )dzdWs ] =

O(|π|).

) L1 L0 u(z, Xz )dWz ds ] =

O(|π|).

) L1 L1 u(z, Xz )dWz dWs ] =

0.

ti

∫t Proof. Observing that H i = h1i tii+1 dWs . By using the Cauchy Schwartz inequality and lemma 5.7.5 in [8], we get ∫ ti+1 ∫ s L0 L0 u(z, Xz )dzds] Eti [Hi ti −1 hi 2

≤

(

ti

∫

ti+1

∫

s

(L0 L0 u(z, Xz ))2 dzds]

E ti [ ti

) 12

3

= O(|π| 2 ).

ti

Let we turn to ii) and iii), we recall that L1 L0 u, L0 L1 u have polynomial growth, by lemma 5.7.2 [8], we have: ∫ ∫ ∫ t ∫ s i+1 ( ti+1 s 0 1 ) Eti [H i L L u(z, Xz )dzdWs + L1 L0 u(z, Xz )dWz ds ] ti

ti

ti

ti

∫ t ∫ s ∫ ti+1 ∫ s i+1 1 = L1 L0 u(z, Xz )dzds] E ti [ L0 L1 u(z, Xz )dzds + hi ti ti ti ti = O(|π|). Finally, by lemma 5.7.1 in [8], we obtain: [ ( ∫ ti+1 ∫ s )] E ti H i L1 L1 u(z, Xz )dWz dWs ti

=

[( 1 E ti hi

∫

ti ti+1

∫

ti

s

)] L1 L1 u(z, Xz )dWz ds = 0.

ti

□ We are now ready to state our first result, which provides an error estimate of the approximation scheme Theorem 3.6. Under (Hr ), we get: Γ(π) = O(|π|2 ).

(3.8)

28

HANI ABIDI AND HABIB OUERDIANE

Proof. We recall that: Y¯ti = Z¯t = i

Eti [Yti+1 + hi f (ti , Xi+1 , Yti+1 , Zti+1 )] Eti [H i Yti+1 ].

For the part of Z, we have Z¯ti = Eti [H i Yti +1 ] = Eti [H i u(ti+1 , Xti+1 )] [ ( )] = Eti H i u(ti , Xti ) + hi L0 u(ti , Xti ) + L1 u(ti , Xti )(Wti+1 − Wti ) + Ri ∫ ∫ ( ti+1 s 0 0 = L(1) u(ti , Xti ) + Eti [H i L L u(z, Xz )dzds ∫

ti+1

∫

ti ti+1

∫

ti

∫

ti+1

∫

s

L1 L0 u(z, Xz )dWz ds +

+ ∫

ti

s

ti s

L0 L1 u(z, Xz )dzdWs ti

ti

)

L1 L1 u(z, Xz )dWz dWs ].

+ ti

ti

By lemma 3.5, we get Z¯ti = Zti + Oti (|π|). Indeed, by the truncation error definition for the Z component, we have: ΓZ (π) = O(|π|2 ). Now the error for the Y-part is: [ ] Y¯ti = Eti Yti+1 + hi (f (ti+1 , Xti+1 , Yti+1 , Zti+1 ) [ ] = Eti u(ti+1 , Xti+1 ) − hi L0 u(ti+1 , Xti+1 ) = Yti + O(|π|2 ). So ΓY (π) = O(|π|2 ). □ Theorem 3.7. Under hypothesis Hr : ε(Y, π) + ε(Z, π) ≤ C|π|2 where C is a positive random variable. Proof. Now, we consider that (Y, Z) is a solution of perturbed scheme, setting ξiY = Yti − Y ti and ξiZ = Zti − Z ti . By Theorem 3.6, we get thus n−1 ∑ i=0

hi E[

1 Y 2 |ξ | + |ξiZ |2 ] ≤ C|π|2 . h2i i

So we can conclude by stability error given by theorem 3.4, that: ε(Y, π) + ε(Z, π) ≤ C|π|2 .

BSDE AND RBSDE

29

□ 3.3. Error of implementable scheme for FBSDE. In this section, we are interested to the error due to the numerical illustration of the result presented above. Firstly, by the approximation of multiple Itˆ o integral [8] p 225, let we b and defined by: consider a weak Taylor approximation of X denoted by X bi+1 X

bi + L0 (X bi )hi + L1 (X bi )∆W ct = X (3.9) i ( ) 1 bi ) (∆W ct )2 − hi + 1 L0 L0 (X bi )h2i + L1 L1 (X i 2 2 1 bi )hi ∆W ct . + [L1 L0 + L0 L1 ](X i 2 b t [.] the related conditional bt ) and E We denote by (Fbt ) the filtration generated by (X expectation. The empirical scheme we use in practice is the following: (Ybn , Zbn ) Ybi Zbi

bt ), L1 u(tn , X bt )) (u(tn , X n n b t [Ybi+1 + hi f (ti , X bi+1 , Ybi+1 , Zbi+1 )] = E i b b i Ybi+1 ]. = Et [H

=

i

ct ∆W i hi

bi = . where H Let now give a lemma to illustrate our convergence result. Lemma 3.8. For u smooth enough bt )] = i)Eti [u(ti+1 , X i+1 i b ii)Eti [H u(ti+1 , Xti+1 )] =

bt ) + hi L0 u(ti , X bt ) + O(h2i ). u(ti , X i i 1 b L u(ti , Xt ) + O(hi ). i

(3.10) (3.11)

Proof. For ii) it is the same proof like Theorem 3.6 in a discrete time. Now we pass to prove i), For v smooth enough and by lemma 5.7.1 in [8], we get: Eti [u(ti+1 , Xti+1 )] =

u(ti , Xti ) + L0 v(ti , Xti )hi + L0 L0 u(ti , Xti )

h2i + O(h3i ). 2 □

b is given by 3.9, then Theorem 3.9. If we assume (Hr ) is verified and X Y0 − Yb0 = O(h2i ). bt ), Zei = L1 u(ti , X bt ) and Proof. Let us define Yei = u(ti , X i i Y˘i Z˘i

bt [Yei+1 + hi f (ti+1 , X bi+1 , Yei+1 , Zei+1 )] = E i bt [H b i Yei+1 ]. = E i

Analog to the proof of theorem 3.6, by using lemma 3.8, we compute that: For the Y Part, [ ] b t u(ti+1 , X bt ) − hi L0 u(ti+1 , X bt ) Y˘ti = E i i+1 i+1 = Yei + O(|π|2 ).

30

HANI ABIDI AND HABIB OUERDIANE

For the Z part, Z˘ti

=

b t [H i Yt +1 ] E i i i b Et [H u(ti+1 , Xt

=

Zei + O(|π|).

=

i

i+1

)]

b is the solution of perturbed scheme (Ye , Z) e it follows that ξ Y = Yb − Y˘ If (Yb , Z) ˘ and by proposition 3.4 and ξ Z = Zb − Z, Y0 − Yb0 = O(|π|2 ). □ Now, we explicit the description of tree method combining weak Taylor scheme for the forward process with binomial approximation of Brownian motion. b then we introduce a Firstly, we describe how to obtain the tree associated to X, binomial approximation of brownian motion, given by, for i ≤ n − 1, √ √ ct − W ct = − hi ) = 1 . ct − W ct = hi ) = P (W P (W i+1 i i+1 i 2 ct , we see that each node of X bi will give two new nodes of ∆√W From the definition i √ b b b Xi , h , X b Xi ,− h depending on the value of the value taken by ∆W ct . denoted X i i+1 i+1 Secondly, we turn to the approximation of backward component, which is combi . We observe that the scheme can be writen puted along the tree built for X (Ybn , Zbn ) Ybi Zbi

bt ), L1 u(tn , X bt )) (u(tn , X n n b t [Ybi+1 + hi f (ti , X bi+1 , Ybi+1 , Zbi+1 )](X bi ) = E i i b t [H bi ). b Ybi+1 ](X = E

=

i

bi using the quadrature rule: The above quantitie can be computed for each node X √ √ b t [φ(X bi+1 )](X bi ) = 1 [φ(X b Xbi , h )] + 1 [φ(X b Xbi ,− h )]. E i i+1 i+1 2 2 b b b t [H b b b i Ybi+1 ] Finally, we need to compute Eti [Yi+1 ], Eti [f (ti , Xi+1 , Ybi+1 , Zbi+1 )] and E i using the same quadrature rule. 4. Implementable Scheme for FBSDE To illustrate our previous results, we will focus on the simple case where X = W. So this step is based on the following BSDE ∫ T ∫ T Yt = g(WT ) + f (s, Ws , Ys , Zs )ds − Zs dWs . t

t

For i < n, the BTZ scheme can be approximated numerically by: { i ,x Yi (x) = Eti [Yi+1 (Wtti+1 )] + hi f (ti , x, Yi (x), Zi (x)) t ,x

i ,x Zi (x) = Eti [Yi+1 (Wtti+1 )

where

,x Wttii+1

Wt i

i+1

hi

is a Brownian motion started by x in ti .

−x

]

(4.1)

BSDE AND RBSDE

31

For this convergence method, we approximate the Brownian motion by a random walk. The Donsker Theorem gives an approximation for Brownian motion as follows, choosing a large integer k > 1, tk 1 ∑ i √ Wk = ξ k i=1

and ξ is a random walk. In our paper, we are interested by the ξ i defined by: 1 P(ξ 1 = 1) = P(ξ 2 = −1) = . 2 c We denote by W the approximation of W , and we define recursively : √ ct − W ct = hξ i . W i+1 i c0 = 0. with W In the following, we describe how to obtain a reasonable scheme for (Y,Z) which is given by Chassagneux and Dan Crisan [3] for the Runge Kutta scheme. The convergence analysis is based on the stability of the scheme and some kinds of truncation error. We are particularly interested by the initial value of Y , which is deterministic but we will need to measure the error at each date to obtain the error at the beginning. So for the Y, a sensible choice is ε(Y, π) = sup0≤i≤n E[|Yti − Ybi |2 ]. The error for the process Z here is given by this following structure ∫ T ε(Z, π) = E[ |Zti − Zbi |2 ds]. 0

Notations : First, we need to consider ’functional’ version of the schemes above. Let us introduce the following operator, related to the theoretical schemes i ,x RiZ [φY , φZ ](x) = E[H i φY (Wtti+1 )] i ,x i ,x i ,x i ,x RiY [φY , φZ ](x) = E[φ(Wtti+1 ) + hi f (ti+1 , Wtti+1 , φY (Wtti+1 ), φZ (Wtti+1 ))].

Similarly, let us define for the fully discrete scheme-operators bH biZ [φY , φZ ](x) = E[ b i φY ( W ctti ,x )] R i+1 b W biY [φY , φZ ](x) = E[φ( ctti ,x ) + hi f (ti+1 , W ctti ,x , φY (W ctti ,x ), φZ (W ctti ,x ))]. R i+1 i+1 i+1 i+1 Using the B-T-Z scheme, we give the functional version as follows: { ybi (x) = RiY [b yi+1 , zbi+1 ](x) zbi (x) = RiZ [b yi+1 , zbi+1 ](x) given initial data (b yn , zbn ) = (u(tn , .), ∂x u(tn , .)). Due to the Markov property of ct0,x ), it is easily checked that the discrete process (W ct0,x ) and Zbi = zbi (W ct0,x ). Ybi = ybi (W

32

HANI ABIDI AND HABIB OUERDIANE

Finally, we define: ct0,x ) and Zei = ∂x u(ti , W ct0,x ). Yei = u(ti , W i i Observe that Ye0 = u(0, x) and that, (Ybn , Zbn ) = (Yen , Zen ). Stability : The key observation is that (Yei , Zei ) can be seen as a perturbed version { e +s ξ Y Yei = Eti [Yei+1 + hi f (ti+1 , Xi+1 , Yei+1 , Z)] i Zei = Eti [H i Yei+1 ] +s ξiZ where the local error due to the ’space-discretization’ is: { s Y bY )[u(ti+1 , .), ∂x u(ti+1 , .)](W ct0,x ) ξi = (RiY − R i i s Z Z Z b )[u(ti+1 , ∂x u(ti+1 , .)](W ct0,x ) ξi = (Ri − R i i Now, we can conclude the error with computing the next Theorem. Theorem 4.1. Under (Hr ), If f is a Lipshitz function, then E[|δY0 |2 ] + E[|δZ0 |2 ] ≤ c|π|2 .

(4.2)

where δY0 = Y0 − Ye0 and δZ0 = Z0 − Ze0 . Proof. In a first time, we are interested by the convergence of Y. This proof use the Markovian property of Brownian motion for changing the conditional expectation considered difficult to approximate . Then, we approximate the truncation error using the Itˆ o-Taylor expansion. So we can define the local error s ξiY : s Y ξi

biY [φY , φZ ](x) = E[φY (Wtti ,x )] − E[φY (W ctti ,x )] = RiY − R i+1 i+1 [ ti ,x ti ,x ti ,x Y Z +hi (E f (ti+1 , Wti+1 , φ (Wti+1 ), φ (Wti+1 ))] [ ] ctti ,x , φY (W ctti ,x ), φZ (W ctti ,x )) ). −E f (ti+1 , W i+1

i+1

i+1

ct = W ct − W ct , from the Taylor expansion, we have: Let ∆W i i+1 i √ t ,x ct i ) = φY (x + hi ξ i ) φY (W i+1 √ ′ 1 = φY (x) + (φY ) (x) hi ξ i + (φY )(2) (x)hi (ξ i )2 2 3 1 Y (3) 1 Y (4) i 3 2 + (φ ) (x)hi (ξ ) + (φ ) (x)h2i (ξ i )4 + O(h4i ). 3! 4! So, we get ctti ,x )] = E[φY (W i+1

1 1 φY (x) + (φY )(2) (x)hi + (φY )(4) (x)h2i + O(h3i ). 2 4!

BSDE AND RBSDE

33

Let ∆Wti = Wti+1 − Wti , using the Itˆ o-Taylor expansion, we have: i ,x E[φY (Wtti+1 )] = E[φY (Wttii ,x + ∆Wti )] [ h2 = E φY (x) + L(0) φY (x)hi + L(1) φZ (x)∆Wti + L(0) L(0) φY (x) i 2 ∫ ti+1 ∫ s ∫ ti+1 ∫ s +L(1) L(1) φY (x) dWz dWs + L(1) L(0) φY (x) dzdWs

ti ti+1

∫

ti s

∫

+L(0) L(1) φY (x) ti

ti

] dWz ds .

ti

ti

Then, we obtain i ,x ctti ,x )] = O(h2i ). E[φY (Wtti+1 ) − φY (W i+1

Under the lipschitz assumption of f, we remark that: bY )[φY , φZ ] = O(h2 ). (RiY − R i i Using the matching moment property of W ti ,x , we easily obtain that [ ] biY [φY , φZ ]|2 (Wt0,x ) ≤ C|π|4 . E |RiY [φY , φZ ] − R i Now, by using the binary random walk to approximate the Brownian motion, we can see that: c ti ,x − x W ′ 1 ctti ,x ) ti+1 ] = (φY ) (x) + (φY )(3) (x)hi + O(h2i ). E[φY (W i+1 hi 3! On the other hand: ∆Wti ∆Wti i ,x E[φY (Wtti+1 ) ] = E[φY (Wttii ,x + ∆Wti ) ] hi hi [( = E φY (x) + L(0) φY (x)hi + L(1) φY (x)∆W ∫ t∫ s h2i (0) (0) Y (1) (1) Y +L L φ (x) + L L φ (x) dWz dWs 2 t0 t0 ∫ t∫ s ∫ t∫ +L(1) L(0) φY (x) dWz ds + L(0) L(1) φY (x) t0

t0

t0

s

t0

dzdWs

) ∆W ] . hi

We get the following result: |s ξiZ | = O(hi ). Since f is a Lipshitz-function, then the B-T-Z scheme (3.3) is L2 -stable. So, we obtain: E[|δY0 |2 ] ≤ (

n−1 ∑

hi E[

i=0

1 s Y 2 (| ξi | ) + |s ξiZ |2 ]) h2i

= O(h2i ). □

34

HANI ABIDI AND HABIB OUERDIANE

Remark 4.2. We remark that if ∫ t ∫ t Xt = a(s, Xs )ds + b(s, Xs )dWs 0

(4.3)

0

∫

∫

T

T

f (s, Xs , Ys , Zs )ds −

Yt = g(XT ) +

Zs dWs .

t

t

Under some assumptions, we can use the Girsanov Theorem [8] and the martingal representation to change Xt given in (4.3) to ft dXt = c(Xt )dW e and c ∈ L2 (Ft ). After that, we can ft is a Brownian motion respect to P where W use the same idea to get the convergence of scheme. 4.1. Example of numerical simulation: In this subsection, we focus on the problem of the simulation of the B-T-Z scheme approximation. This algorithm is based on the approximation of Brownian motion by a binary random walk. Then we approximate the conditional expectation of Euler scheme in a backward direction. We consider the process (Xt , Yt , Zt ) = (Wt ,

exp(−Wt − 4t ) 1 , ). 1 + exp(−Wt − 4t ) (1 + exp(−Wt − 4t ))2

This process is a solution of the (decoupled) FBSDE Xt Yt

= Wt

∫

(4.4)

∫

T

T

f (Ys , Zs )ds −

= g(XT ) +

Zs dWs

t

(4.5)

t

where the driver f is given by 3 f (y) = −z( − y) 4 and gT (x) = Let h =

1 n,

1 1 + exp(−x −

T 4

)

.

this process Y and Z defined in (4.5) are approximated as follows: { i ,x yi (x) = Eti [yi+1 (Wtti+1 )] + hi f (yi (x), zi (x)) t ,x (4.6) Wt i −x i ,x zi (x) = Eti [yi+1 (Wtti+1 ) i+1 ] hi

,x where Wttii+1 is a Brownian motion started by x in ti . Using the C++, we test the error between the Euler scheme and the exact solution for the initial value Y0 and Z0 . Table 1 and 2 shows the decrease of Y and Z errors, if we reduce the step discretization h. We also note that the slope of the two curves are equal to -1.

BSDE AND RBSDE

log(n) log(error of Y ) 1.3862 −6.189 1.791 −6.561 2.3025 −7.045 2.564 −7.298 2.708 −7.438 tab.1.error for the Y-part

35

log(n) log(error of Z) 1.386 −4.575 1.791 −4.938 2.079 −5.205 2.302 −5.415 2.708 −5.804 tab.2.error for the Z-part

5. Reflected BSDE’s With Continuous Barrier In this section, we gives some estimates for the extension of B-T-Z scheme for reflected FBSDE ∫ t ∫ t Xt = X0 + b(s, Xs )ds + σ(s, Xs )dWs 0

∫

0

T

∫

T

f (Ys , Zs )ds + AT − At −

Yt = g(XT ) + t

Zs dWs t

where At is an increased continuous process, whose roles are to keep the process Y up to the continuous lower barrier L : ∫ T (Ys − Ls )dAs = 0. 0

5.1. Penalization approximation. We consider here the following sequence of BSDE’s ∫ T ∫ T ∫ T m m m m b b (l(Xs ) − Ys )+ ds − Zsm dWs Yt = g(XT ) + f (Ys , Zs )ds + m t

t

t

where m −→ ∞. If we use an explicit B-T-Z-scheme, this would lead to the following method: Yiπ,m

π,m π,m bi ) − Y π,m )+ ]. = Eti [Yi+1 + hi f (Yi+1 , Ziπ,m ) + mhi (l(X i+1

Ziπ,m

π,m = Eti [H i Yi+1 ].

In the next part, we are interested by the optimal case, where m =

n T.

5.2. General a priori estimates. The two results which will be showed, establish the stability estimations for a class of RBSDE. To estimate the discrete RBSDE, let’s introduce an extension of B-T-Z scheme: Yj,N hi Zj,i

= g(XT,j ), Yj,i = Ei [Yj,i+1 + fj,i (Yj,i+1 , Zj,i )hi + (lj (Xi ) − Yj,i+1 )+ ], = Ei [Yj,i+1 ∆Wi ],

where i ∈ {0, ..., N − 1}, j ∈ {1, 2}. To establish the stability result, the lemma below, is an intermediate result to give the global estimations. Denote by ∆ξ ∆fi e ∆l(Xi )+

= g(XT,1 ) − g(XT,2 ), ∆Yi = Y1,i − Y2,i , = f1 (Y1,i+1 , Z1,i ) − f2 (Y2,i+1 − Z2,i ) = (l1 (Xi ) − Y1,i+1 )+ − (l2 (Xi ) − Y2,i+1 )+ .

36

HANI ABIDI AND HABIB OUERDIANE

Lemma 5.1. (Local estimates). For j ∈ {1, 2}, assume that g(XT,j ) is in L2 (FT ). for each i ∈ {0, ..., N }, assume that f1,i (Y1,i+1 , Z1,i ) is in L2 (FT ) and f2,i (y, z) is Lipshitz continuous w.r.t y and z, with a finite lipschitz constant Lf2 ,i ≥ 0. Then, for any hi ≤ T and γi ≥ 0 satisfying 6(hi + γ1i ) ≤ 1, if follows that |∆Yi |2

≤

1 1 (1 + (γi + )hi )Ei (|∆Yi+1 |2 ) + 3(hi + )hi Ei [|∆fi |2 ] 2 γi +(1 + γi hi )Ei [∆l(Xi )]2 .

(5.1) (5.2)

Proof. Preliminary estimates for ∆Zi : Since the Brownian motion is conditionally centered, it follows that: hi ∆Zi = Ei [(∆Yi+1 − Ei [∆Yi+1 ])∆Wi ]. By the cauchy-Schwartz inequality, we get: 1( |∆Zi |2 ≤ Ei [(∆Yi+1 )2 ] − (Ei [∆Yi+1 ])2 ) hi Let: Yj,i = Ei [Yj,i+1 + fj,i (Yj,i+1 , Zj,i )hi + (lj (Xi ) − Yj,i+1 )+ ]. We get: ∆Yi

∆Yi

[ ] = Ei ∆Yi+1 + hi [f1 (Y1,i+1 , Z1,i ) − f2 (Y2,i+1 , Z2,i )] + ∆e l(Xi ) [ = Ei ∆Yi+1 + hi [f1 (Y1,i+1 , Z1,i ) − f2 (Y1,i+1 , Z1,i ) ] +f2 (Y1,i+1 , Z1,i ) − f2 (Y2,i+1 , Z2,i )] + ∆e l(Xi )

[ ] = Ei ∆Yi+1 + hi [∆fi + f2 (Y1,i+1 , Z1,i ) − f2 (Y2,i+1 , Z2,i )] + ∆e l(Xi ) .

We denote now by: ∆Gi = ∆fi + f2 (Y1,i+1 , Z1,i ) − f2 (Y2,i+1 , Z2,i ). Therefore, we can distinguish four cases: I- If l1 (Xi ) ≤ Y1,i+1 and l2 (Xi ) ≤ Y2,i+1 [ ] ∆Yi = Ei ∆Yi+1 + hi ∆Gi . II-If l1 (Xi ) ≤ Y1,i+1 and l2 (Xi ) ≥ Y2,i+1 [ ] ∆Yi = Ei ∆Yi+1 + hi ∆Gi + Y2,i+1 − l2 (Xi ) [ ] ≤ Ei Y1,i+1 − l2 (Xi ) + hi ∆Gi [ ] ≤ Ei ∆Yi+1 + hi ∆Gi .

BSDE AND RBSDE

37

II- If l1 (Xi ) ≥ Y1,i+1 and l2 (Xi ) ≤ Y2,i+1 [ ] ∆Yi = Ei ∆Yi+1 + hi ∆Gi − Y1,i+1 + l1 (Xi ) [ ] = Ei − Y2,i+1 + l1 (Xi ) + hi ∆Gi [ ] ≤ Ei ∆l(Xi ) + hi ∆Gi . II-If l1 (Xi ) ≥ Y1,i+1 and l2 (Xi ) ≥ Y2,i+1 [ ] ∆Yi = Ei ∆Yi+1 + hi ∆Gi − Y1,i+1 + l1 (Xi ) + Y2,i+1 − l2 (Xi ) [ ] = Ei ∆l(Xi ) + hi ∆Gi . Estimates for ∆Yi . We distinguish two cases. First case: let l1 (Xi ) ≤ Y1,i+1 , We have ∆Yi = Ei [∆Yi+1 + hi ∆fi + hi (f2,i (Y1,i+1 , Z1,i ) − f2,i (Y2,i+1 , Z2,i )]. Combining the Young inequality and the lipschitz property of f2,i , we deduce that: (∆Yi )2

≤ (1 + γhi )(Ei [∆Yi+1 ])2 ] 1 [ +3(hi + )hi Ei [|∆fi |2 ] + L2f2 ,i Ei [(∆Yi+1 )2 ] + L2f2 ,i |∆Zi |2 γ 1 1 ≤ (1 + γhi − 3L2f2 ,i (hi + ))(Ei [∆Yi+1 ])2 + 3(hi + )hi Ei (|∆fi |2 ) γ γ ] [ 1 1 + 3(hi + )hi L2f2 ,i + 3L2f2 ,i (hi + ) Ei [(∆Yi+1 )2 ]. γ γ

The assumption on γi and hi ensure that 1 + γi hi − 3L2f2 ,i (hi + γ1i ) ≥ 0 for any hi , whence, applying Jensen’s inequality to the term in (Ei ∆Yi+1 )2 , it follows that: [ 1 1 (∆Yi )2 ≤ (1 + γi hi − 3L2f2 ,i (hi + ))(Ei [(∆Yi+1 )2 ]) + 3(hi + )hi Ei [|∆fi |2 ] γi γi [ ] 1 1 2 + 3(hi + )hi L2f2 ,i + 3L2f2 ,i (hi + ) Ei [∆Yi+1 ] γi γi 1 1 2 ]) + 3(hi + )hi |∆fi |2 ]. = (1 + γi hi + 3L2f2 ,i hi (hi + ))(Ei [∆Yi+1 γi γi Finally, we use 3(hi + γ1i )hi L2f2 ,i ≤ h2i to complete the first case. Second case: let l1 (Xi ) ≥ Y1,i+1 . We have ∆Yi = Ei [∆l(Xi ) + hi ∆fi + hi (f2 (Y1,i+1 , Z1,i ) − f2 (Y2,i+1 , Z2,i ))], so we get: (∆Yi )2

≤

(1 + γi hi )(Ei [∆l(Xi )])2 ] [ 1 +3(hi + )hi Ei [|∆fi |2 ] + L2f2 ,i Ei [(∆Yi+1 )2 ] + L2f2 ,i |∆Zi |2 γi

38

HANI ABIDI AND HABIB OUERDIANE

] 1 [ ) E[(∆Yi+1 )2 ] − (Ei [∆Yi+1 ])2 γi

≤ 3L2f2 ,i (hi +

1 )hi L2f2 ,i Ei [(∆Yi+1 )2 ] γi 1 +3(hi + )Ei [|∆fi |2 ] + Kl (1 + γi hi )(Ei [∆l(Xi )])2 . γi

+3(hi +

The application of 6(hi + (∆Yi )2

1 γi )

≤ 1, gives:

≤ 3L2f2 ,i (hi +

1 )(hi + 2)Ei [(∆Yi+1 )2 ] γi

1 )hi [Ei [|∆fi |2 ] + (1 + γi hi )(Ei [∆l(Xi )])2 . γi hi 1 ≤ (1 + )Ei [(∆Yi+1 )2 ] + 3(hi + )hi [Ei [|∆fi |2 ] 2 γi +(1 + γi hi )(Ei [∆l(Xi )])2 . +3(hi +

□ Proposition 5.2. (Global pointwise estimates). For j ∈ {1, 2}, assume that ξj is in L2 (FT ). Moreover, for each i ∈ {0, ..., N − 1}, assume that f1,i (Y1,i+1 , Z1,i ) is in L2 (FT ) and f2,i (y, z) is Lipschitz continuous w.r.t y and z, with a finite Lipschitz constant Lf2,i ≥ 0. Then, for any time grid π and γ ∈ (0, ∞)N satisfying 6(hi + γ1k )L2f2,k ≤ 1 for all k ≤ N − 1, we have for 0 ≤ i ≤ N : |∆Yi |2 Γi

+

N −1 ∑

hi Ei (|∆Zk |2 )Γk

k=i

≤

N −1 ( ) ∑ 1 C ΓN Ei (|∆ξ|2 ) + 3 ( + hi )hi Ei [|∆fk |2 ]Γk γk k=i

+6 where Γi =

∏i−1

k=0 (1

N −1 ∑

(1 + γi hi )(Ei [∆l(Xi )])2 ,

k=i

+ γk hk ) and C = (1 + T )eT /2 .

Proof. Let λi = (1 + (γi−1 + 21 )λi−1 ), where λ0 = 1. Multiplying the both sides of the equation by λi , we obtain 1 |∆Yi |2 λi ≤ Ei [|∆Yi+1 |2 ]λi+1 + 3(hi + )hi Ei [|∆fi |2 ]λi + (1 + γi hi )Ei [∆l(Xi )]2 λi , γi summing both sides of the inequality from i to N-1 and taking the conditional expectation Ei [.], we deduce that: (∆Yi )2 λi

≤ λN Ei (|∆ξ|2 ) + 3

N −1 ∑ k=i

×hk Ei [|∆fk |2 ]λk +

(

1 + hk ) γk

N −1 ∑

(1 + γk hk )Ei [∆l(Xk )]2 λk .

k=i

BSDE AND RBSDE

From the simple inequality Γi ≤ λi = exp( follows that, for all i ∈ {0, ..., N }, (∆Yi )2 Γi

∑i

ln(1 + (γk + 21 )hi )) ≤ eT /2 Γi , it

k=0

≤ eT /2 ΓN Ei (|∆ξ|2 ) + 3eT /2

39

N −1 ∑ k=i

+

N −1 ∑

(

1 + hh )hk Ei [|∆fk |2 ]Γk γk

(1 + γk hk )Ei [∆l(Xk )]2 Γk .

k=i

For the estimation of ∆Zi , we remark that above by: N −1 ∑

∑N −1 k=i

hk Ei [|∆Zk |2 ]Γk is bounded

Γk+1 (Ei [(∆Yk+1 )2 ] − Ei [Ek (∆Yk+1 )2 ])

k=i N −1 ∑

≤ ΓN Ei [|∆ξ| ] + 2

Γk (Ei [(∆Yk )2 ] − (1 + γk hk )Ei [Ek (∆Yk+1 )2 ]).

k=i+1

Therefore, we can distinguish two cases: If l1 (Xi ) ≤ Y1,i+1 , we can see that Ei [(∆Yk )2 ] − (1 + γk hk )Ei [Ek (∆Yk+1 )2 ] is bounded above by: [ ] 1 3( + hk )hk Ei [|∆fk |2 ] + L2f2,k Ei [(∆Yk+1 )2 ] + L2f2,k Ei [|∆Zk |2 ] . γk By using the Jensen Inequality and the statement 6(hi + ∑N −1 that k=i hk Ei [|∆Zk |2 ]Γk is bounded above by: N −1 ∑

ΓN Ei (∆ξ 2 ) + 3

k=i+1

+ 3

N −1 ∑

(

k=i+1

+ 3

N −1 ∑

(

k=i+1

(

1 2 γk )Lf2,k

≤ 1, it follows

1 + hk )hk L2f2,k Ei [|∆Zk |2 ]Γk γk

1 + hk )hk Ei (|∆fk |2 )Γk γk 1 + hk )hk L2f2,k Ei [(∆Yk+1 )2 ]Γk . γk

Using the assumptions on γk and ∆k of the proposition statement, it follows that, ∑N −1 for the first case, k=i hi Ei [|∆Zk |2 ]Γk is bounded by: 2ΓN Ei [|∆ξ 2 |] + 6

N −1 ∑ k=i+1

+

N −1 ∑

(

1 + hk )hk Ei (|∆fk |2 )Γk γk

hk Ei [(∆Yk+1 )2 ]Γk

k=i+1

≤ (2 + T eT /2 )ΓN Ei [|∆ξ 2 |] + (6 + 3T eT /2 )

N −1 ∑ k=i+1

(

1 + hk )hk Ei (|∆fk |2 )Γk . γk

40

HANI ABIDI AND HABIB OUERDIANE

where the estimate on ∆Y is used in the last inequality. I- If l1 (Xi ) ≥ Y1,i+1 , we can see that (Ei [(∆Yk )2 ] − (1 + γk hk )Ei [Ek (∆Yk+1 )2 ]) is bounded above by: [ 1 (1 + γi hi )Ei [∆l(Xi )]2 + 3( + hk )hk Ei [|∆fk |2 ] γk ] +L2f2,k Ei [(∆Yk+1 )2 ] + L2f2,k Ei [|∆Zk |2 ] − (1 + γk hk )Ei [Ek (∆Yk+1 )2 ]). Using the Jensen Inequality and the statement 6(hi + γ1k )L2f2,k ≤ 1, it follows that ∑N −1 2 k=i hk Ei [|∆Zk | ]Γk is bounded above by: ΓN Ei (∆ξ 2 ) + 3

N −1 ∑

(

k=i+1

+3

N −1 ∑

(

k=i+1

+

1 + hk )hk L2f2,k Ei [|∆Zk |2 ]Γk γk

1 + hk )hk Ei (|∆fk |2 )Γk γk

N −1 N −1 ∑ 1 ∑ Ei [(∆Yk+1 )2 ]Γk + 3 (1 + γi hi )Ei [∆l(Xi )]2 Γk . 2 k=i+1

k=i+1

Therefore: 2ΓN Ei [|∆ξ 2 |] + 6

N −1 ∑ k=i+1

+

N −1 ∑

(

1 + hk )hk Ei (|∆fk |2 )Γk γk N −1 ∑

hk Ei [(∆Yk+1 )2 ]Γk + 6

(1 + γi hi )Ei [∆l(Xi )]2 Γk

k=i+1

k=i+1

≤ (2 + T eT /2 )ΓN Ei [|∆ξ 2 |] + (6 + 3T eT /2 )

N −1 ∑ k=i+1

) ×hk Ei (|∆fk |2 Γk + 6

N −1 ∑

(1 + hk ) γk

(1 + γi hi )Ei [∆l(Xi )]2 Γk .

k=i+1

□

References 1. Bouchard, B and Touzi, N.: Discrete-Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations. Stochastic Processes and their Applications, 111 (2004) 175–206. 2. Briand, P. Delyon, B and Memin, J.: Donsker theorem for BSDEs, Electronic Comuunication in Probability 6 (2001) 1–14. 3. Chassagneux, J. and Crisan, D.: Runge-Kutta schemes for BSDEs, Annals of Applied Probability, 24 (2) (2014) 679–720. 4. Cvitanic, J. and Karatzas, I.: Backward SDEs with reflection and Dynkin games, Ann. Probab. 24 (4)(1996) 2024–2056. 5. El Karoui, N. Peng, S. and Quenez, M.C.: Backward Stochastic Differential Equation in finance, Mathematical finance, 7 (1) (1997) 1–71.

BSDE AND RBSDE

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6. El Karoui, N. Kapoudjian, C. Pardoux, E. Peng, S and Quenez, M.C.: Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 (1997) 702–737. 7. Gobet, E. and Labart, C.: Error expansion for the discretization of bacward stochastic differential equations, Stochastic Processes and their Applications, 117 (2007) 803–829. 8. Kloeden, P.E and Platen, E.: Numerical solutions of Stochastic Differential Equations, Applied Math. 23, Springer, Berlin, 1992. 9. Ninomiya, S. and Victoir, N.: Weak Approximations of Stochastic Differential Equations and Application to Derivative Pricing, Applied mathematical finance, 15 (2008) 107–121. 10. Pardoux, E. and Peng, S.: Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 14 (1990) 55–61. 11. Pardoux, E. and Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic partial differential equations and their applications, Springer, Berlin, 176 (1992) 200–217. 12. Zhang, J.: A numerical scheme for backward stochastic differential equation, Annals of Applied Probability, 14 (2004) 459–488. Hani Abidi: Department of Mathematics, Faculty of Sciences of Tunis University of Tunis El-Manar, 1060 Tunis, Tunisia E-mail address: [email protected] Habib Ouerdiane: Department of Mathematics, Faculty of Sciences of Tunis University of Tunis El-Manar, 1060 Tunis, Tunisia E-mail address: [email protected]