A New Second Order Numerical Scheme for Solving Forward Backward Stochastic Differential Equations with Jumps

http://dx.doi.org/10.4236/am.2016.712121

A New Second Order Numerical Scheme for

Solving Forward Backward Stochastic

Differential Equations with Jumps

Hongqiang Zhou, Yang Li, Zhe Wang

College of Science, University of Shanghai for Science and Technology, Shanghai, China

Received 1 April 2016; accepted 26 July 2016; published 29 July 2016

Copyright © 2016 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator f =r t x y

(

, t, t

)

+h t z

( )

t+g t

( )

Γt linearly de-pending on zt. And we theoretically prove that the convergence rates of them are of second order for solving yt and of first order for solving zt and Γt in

p

L norm.

Keywords

Numerical Scheme, Error Estimates, Backward Stochastic Differential Equations

1. Introduction

convergence rate, such as [2][3].

In this paper, we propose a new second order numerical scheme for the solution of forward-backward sto- chastic differential qquations (FBSDE in short) with jumps with the following form

(

)

(

)

(

) (

)

( )

(

)

0 ₀ , d ₀ , d ₀ , , d , d

d d d , d

t t t

t s s s s k

T T T

t T _t s _t s s _t s k

x x b s x s s x W s x e N s e

y g x f s z W U N s z

σ β ₋

 = + + +

 

 = + − −



∫

∫

∫ ∫

∫

∫

∫ ∫









(1)

From [4], we know that the solution

(

x y z_t, _t, ,_t Γ_t

)

can be represented as

(

)

(

) (

)

(

)

(

)

(

)

{

}

( ) ( )

[

)

, , , , ,

, , , , d , 0, ,

t t t x t t

t t t t

y u t x z u t x t x

u t x t x e u t x e e t T ε

σ

β ρ λ

= = ∇

Γ =

_∫

+ − ∈ (2)

where the vector function u t x

( )

, is the classical solution of the following parabolic differential equation (PDE) of the form

( )

(

( )

( ) ( )

[ ]

( )

)

(

)

( )

, , , , , , , , , 0

,

x

u t x f t x u t x u t x t x u t x u T x g x

σ

 + ∇ =

 

= 

 

(3)

where ∇_xu denotes the gradient of u with respect to the space variable x,

( )

( )

( ) ( )

(

( )

)

( )

( )

(

)

( )

( ) ( ) ( )

[ ]

( )

{

(

( )

)

( )

}

( ) ( )

2 , 1

1

, : , , ,

2

, , , , , d ,

, : , , , d .

d

x i j

i j

x

u u

u t x t x u t x b x x t x

t x x

u t x x e u t x u t x x e e

u t x u t x x e u t x e e

ε ε

σσ

β β λ

β ρ λ

∗ =

∂ ∂

= ∇ +

∂ ∂ ∂

+ + − − ∇

= + −

∑

∫

∫





(4)

2. Preliminaries and Notation

Let T be a fixed positive number and

{

{ }

}

0

, , , _t

t T

P _{≤ ≤}

Ω  be a complete,filtered probability space on which is defined a standard Brownian motion W_t, such that

{ }

t ₀≤ ≤t T is the natural filtration of the Brownian motion

t

W and all the P-null sets are augmented to each σ-field _t. Denote by 2

L the set of all _t -adapted and mean-square-integrable processes.

A process

(

x y zt, t, ,t t

)

: 0,

[ ]

T Rm Rm d ×

Γ × Ω → × is called an 2

L -adapted solution of the FBSDE(1) if it’s

{ }

t -adapted and

2

L -integrable, and satisfies (1). Under some standard conditions on the functions f and h, there is a unique adapted random process.

Now we introduce a new probability space: for

{

Λ_t:t≥0

}

is an exponential martingale and satisfies

( )

1

x t_n Λ =t

 , we define

[ ]

n1 n

t t

X = XΛ + 

 

 

  . The random processes n

t t

Λ , it is easy to verify that n

t t

Λ is an exponential martingale.

( )

2

( )

( ) ( )

(

{ }

)

0 ( ) ( ) ( )d d

0 _{0 0}

0

1

exp d d 1 , d e .

2

t

t t g s e e s

t

s k

s t

h s W h s s g s ρ e N s e − ρ π

≤ ≤

∫ ∫

  _{ }

Λ = _ − _{ } + _



∫

∫



∏

∫

  (5)

Let us first introduce the following lemma.

Lemma 1. Given the time partition 0≤ ≤tn tn+₁≤T, X is a t-measurable random variable,and satisfies

[ ]

X < ∞



 .

[ ]

0 1 1

0

, . . n

n

n _n n

n

n n n n

t x

t t

x x

t t t t

X

X X a s

+

+

 Λ 

  _{ }

= = _ Λ _

Λ

 

  (6)

We use the following Itô-Taylor approximation to solve the forward SDEs with jumps

(

)

1 2

1

,

, ,

n n

n n n

n

t t T

X X I_α f_α t X

α +

+

∈

 

where

( )

( )

( )

( )

( )

(

)

, ,

, ,

, 0,

d , 1, = 0

,

d , 1, 1, 2, ,

d , d , 1, 1

n n

l n

n n

n n

l t r

t

j t

r l

t r t

l t r

t

g l v

I g r l j

I g

I g W l j m

I g N e r l j

τ α τ

α _τ

α τ

α

τ α

− −

− ₋

= =

 

⋅ ≥

 

 _{ }



⋅ =

  _

  _{ ⋅  ≥ ∈}

 

 

 _{ ⋅  ≥ = −}



∫

∫

∫ ∫







(8)

and the coefficient function

(

)

( ) ( )

(

) ( )

(

) ( )

1

1 1

, , 0

, , , , , 1, 0,1, , ,

, , , 1, 1

j

e l

f t x l

f t x e L f t x e l j m L f t x e l j

α α

α α

α α −

− −

= 



=_ ≥ ∈

 _{≥ = −}



 (9)

with

(

)

(

)

( )

(

)

( ) ( )

(

)

(

)

( )

(

)

(

)

(

(

)

)

(

)

[ ]

2

0 , ,

1 , 1 1

, 1 1

1

, , , , , , , , , , , ,

2

, , , , , ,

, , , , , , , , , 0, , , .

d d m

i i j l j

i i l

i i l j

d

k i k

i i

d s

e

L f t x e f t x e a t x f t x e b t x b t x f t x e

t x x x

L f t x e b t x f t x e x

L f t x e f t x c t x v e f t x e t T x R e ε

= = =

= −

∂ ∂ ∂

= + +

∂ ∂ ∂ ∂

∂ =

∂

= + − ∈ ∈ ∈

∑

∑∑

∑

(10)

Now we introduce some basic notations.

•

{ }

t _{t s T≤ ≤} : the σ-field generated by the Brownian motion.

• Throughout this paper, we denote by C a generic constant depending only on T, the upper bounds of the de-rivatives of the functions f.

3. Numerical Schemes for Solving BSDE

From the time interval

[ ]

0,T , we introduce the following time partition: 0= < ⋅⋅⋅ <t₀ tN =T, let

1

n n n

t t+ t

∆ = − and

0≤ ≤ −maxn N 1∆ =tn h. According to (1), it’s easy to obtain that for 0≤ ≤n N−1,

(

) ( )

( )

(

)

(

)

1 1 1

1 , , d d d , d .

n n n

n n _{n n n}

t t t

t t _t s s s s _t s s _t s k

y =y ₊ +

∫

+ r s x y +h s z +g s Γ s−

∫

+z W −

∫ ∫

+ _U N s z (11) From (5) and (11),we have

( ) ( )

( )

d d d ,

n n

t t

t t  ρ e g t Nt h t Wt

Λ = Λ _

_∫

 + _

 (12)

(

) ( )

( )

dy_t = −_r t x y, ,_{t t} +h t z_t+g t Γ_t_dt+z W_td _t+

∫

U N_td_t,

 (13)

d d d d d .

n n n n

t t t t

t yt t yt yt t t yt

Λ = Λ + Λ + Λ (14)

From (12), (13) and (14), by applying Itô formula to n

t t yt

Λ , we obtain the equation

(

)

( )

( ) ( )

d , , d d d .

n n n n

t t t t

t yt tr t x yt t t t zt h t yt Wt t Ut yt

ρ

e g t Nt

Λ = −Λ + Λ _ + _ + Λ

_∫

_ + _ 

 (15)

From (15), it is easy to obtain that for 0≤ ≤n N−1,

(

)

( )

( ) ( )

1 1

1 1

1

, , , , , ,

, ,

, , d d

d ,

n n _n n n _n n n

n n n n n n n

n n n _n n _n n

n n

n _{n n}

n n

t t

t x t t x s t x t x s t x t x

t t t _t t s s _t t s s s

t _{s t x t x}

t s s s

t

y y r s x y s z h s y W

U y ρ e g s N

+ +

+ + +

 

= Λ + Λ − Λ +

 

 

− Λ +

 

∫

∫

∫ ∫





(16)

Taking the conditional mathematical expectation

[ ]

n

x t ⋅

(

)

1 1

1

, , , ,

, , d .

n n n _n n n n

n n n n n

n n n n _n n n

t

t x x t t x x s t x t x

t t t t _t t t s s

y + y + r s x y s

+

 

 

= _Λ _+ _Λ _

 

∫

  (17)

Based on (17), we have

(

)

(

)

(

)

1 1

1 1 1 1

1 1

2

, , , 1 ,

1

1

1

, , , , ,

2

n n n n n n

n n n n n n

n n n n n n n n

t x x t t x n t x x t n t x n

t t t t n n t t t n t yj

j

y + y + + t r t x y +r t x y + + R

+ + + + =    

= _Λ _+ ∆ + _Λ _ +

 

∑

  (18)

where

1 ₁

1 1

, 1 , 1 ,

d ,

2 2

n n n

n n

n _{n n n n}

n n n n n n

n

t _{t x t t x t x}

n x t x

y _t t t t n t t t n t

R + r t t +r t r

+

   

=

_∫

_{ }Λ _ − ∆ _{ }Λ _− ∆ (19)

and

(

1

)

(

1

)

1 1 1 1

2 1 1 1 1

, , 1 , , _.

2

n n n n

n n

n n n n n n

n n n n n n n n

t t x t x t t x t x

n x x

y t t t t n t t t t

R + y y + + t + r r + +

+ + + +

   

= _Λ − _+ ∆ _Λ − _

   

  (20)

According to Lemma 1, the equality 1

1 1

,n , n

n n

n n n

n n n n n

t t x t x

x x

t t yt t yt

+

+ +

_Λ ₌  

    

  , we have

(

)

(

1

)

1

1 1

2

, , , 1 ,

1

1

1 1

, , , , ,

2 2

n n n n n n

n n n n

n n n n n n

t x x t x n t x x n t x n

t t t n n t n t n t yj

j

y y t r t x y t r t x y + + R

+ + + + =     = + ∆ + ∆ _{ }+    

∑

 

  (21)

Let

n

t t t

W W W

∆ = − for t_n≤ ≤t t_n+₁. Then ∆Wt is a standard Brownian motion with mean zero and vari-

ance t−t_n. Now multiply (11) by ∆W_t, taking the conditional mathematical expectation n

[ ]

n

x t ⋅

 on both sides of the obtained equation, and using the Itô isometric formula, we deduce

1 1

1 1 1

, , ,

0 n n n n d n n n n n n n d .

n n n n n n

n n

t _{x t x x t x} t _{x t x}

t s t t t t t s

t f W s y W t z s

+ +

+ + +

     

=

_∫

 _ ∆ _ + _ ∆ _−

_∫

 _{ } (22)

From (22) we have,

1 1 1 1 2 , , 1

0 n n n n n ,

n n n n

t x t x

x n

t t t n t zj

j

y + + W t z R

+ +

=

 

=_{ } ∆ _− ∆ +

∑

(23)

where

(

)

1 1 1 1 1

1 1 1

, , ,

1

, ,

2

d d ,

.

n n n

n n

n _n n _{n n}

n n n n

n n

n n

n

n n

n n n n

t _{t x} t _{t x t x}

n x x

z _t t s t _t t s t

t x t x

n x

z t t t t

R f W s z z s

R y y W

+ + + + + + + +       = _ ∆ _ − _{  }− _   = _ − ∆ _  

∫



∫

  (24) Let 1 1 n n

t t n

N ₊ N ₊

λ

t

∆ = ∆ − ∆ _{, similarly multiplying (11) by} 1 n

t

N ₊

∆ _yields

1 1

1 1 1

, , ,

0 n n n n d n n n n n n n d .

n n n n n n

n n

t _{x t x x t x} t _{x t x}

t s t t t t t s

t f N s y N t s

+ +

+ + +

     

=

_∫

_{ } ∆ _ +_{ } ∆ _−

_∫

_{ }Γ _{ (25)}

From (25) we have,

1 1 1 1 2 , , 1

0 n n n n n ,

n n n n

t x t x

x n

t t t n t j

j

y + + N t R

+ + Γ

=

 

=_{ } ∆ _− ∆ Γ +

∑

₍₂₆₎

where

(

)

1 1 1 1 1

1 1 1

, , ,

1

, ,

2

d d ,

.

n n n

n n

n _n n _{n n}

n n n n

n n

n n

n

n n

n n n n

t _{t x} t _{t x t x}

n x x

t s t t s t

t t

t x t x

n x

t t t t

R f N s s

R y y N

+ + + + + + + + Γ Γ      

= _ ∆ _ − _{ }Γ _− Γ _

    = _ − ∆ _

∫



∫

  

 (27)

Based on (21), (23) and (26), for solving the BSDE (1) we propose the following scheme.

Scheme 1. Given

(

0, N, N, N

)

x y z Γ , solve

(

n1, n, n, n

)

(

)

(

)

(

)

1 2

1

1

1

,

1 1 1

1

1

1

, , , (28)

1 1

, , , , , (29)

2 2

, (30)

. (31)

n n

n n

n n

n

n n

n

n n

n n n n

n

t t T

n x n n n x n n

t n n n t n

n x n

n t t

n x n

n t t

x x I f t x y

y y t f t x y t f t x y

t z y W

t y N

α α

α +

+

+

+

∈

+ + +

+

+

+

 = + _ _

 

 _{=  + ∆ + ∆  }

 _{   }



_{∆ =  ∆ }

  



 

∆ Γ = ∆

 _{ }



∑

 



 

 

4. Error Estimates

In this section, we will give the error estimates of Scheme 1 proposed in Section 3. Now we introduce the error

, n n n t x

n n

y t

e =y −y , n,n n t x

n n

z t

e =z −z and n, n n t x

n n

t

e_Γ = Γ − Γ in Lp norm, where

(

t xn, n, t xn,n, t xn, n, t xn,n

)

t t t t

x y z Γ is the solution of the FBSDEs (1), and

(

n 1, n, n, n

)

x+ y z Γ is the solution of Scheme 1. For the sake of simplicity, we only consider one-dimensional BSDEs

(

i e m. ., = =d 1

)

. However, all error estimate that we obtain in the sequel also hold for general multidimensional BSDEs. In our error analysis, we will use a constraint on the time partition step ∆t_n:

0 1 0 0 1

max

. min

n n N

n n N

t c t

≤ ≤ − ≤ ≤ −

∆ ≤

∆ (32) Let us introduce the following Lemma, its proof can be found in the reference [2].

Lemma 2. Let Rn_yj, R_zjn and RΓn_j (j=1, 2) be the truncation errors defined in (21), (23) and (26), respec- tively. It holds that

{

}

( )

3

1 1 1

0max1 , , ,

n n n

y z

n N R R RΓ C t

≤ ≤ − ≤ ∆ (33)

{

}

( )

3

2 2 2

0max1 , , .

n n n

y z

n N R R RΓ C t

≤ ≤ − ≤ ∆ (34) Here C is a positive constant depending on T. We first give the error estimate for n,n

n

t x n

t

y −y in the following theorem.

Theorem 1. Let

(

, , , ,

)

0

, , ,

n n n n

n n n n

t x t x t x t x

t t t t

t T x y z

≤ ≤

Γ (0≤ ≤n N−1) be the solution of the FBSDE (1) and

(

1

)

, , ,

n n n n

x+ y z Γ be the solution of Scheme 1. Assume yN =

ϕ

( )

XN . Then for sufficiently small time step ∆t_n, we have

, 2

0max

n n n

p

t x n p

t

n N y y Ch

≤ ≤

 ₋ _≤

 

 

 (35)

for 1≤ ≤ ∞p , where C is a constant depending on T. Proof. Let n, n

n t x

n n

y t

e =y −y . Subtracting (29) from (21) to get

(

)

1

2

, 1 1

=1

1

. 2

n

n n

n

n n n

t x

n x n n x n n

y t t n f t f yj

j

e = _y ₊ −y +_+ ∆t e + _e+_ +

∑

R (36) Under the conditions of the theorem and by Lemma 2,we deduce that,

( )

(

)

( )

3

1 1

3 1

1 1

2 2

1 ,

n n

n n

n n

n x n n x n

y t y n y n t y n

x n n

n t y n y n

e e t L e t L e C t

C t e C t e C t

+ +

+

 

 

≤ _{ }+ ∆ + ∆ _{ }+ ∆

 

≤ + ∆ _{ }+ ∆ + ∆

 



 



(37)

where L is the Lipschitz constant of f t x y

(

, ,

)

with respect to y. Applying the inequality

(

)

(

( )

1

)

1

1

2 1

1 2 1

p

p _{p p p}

p

a b a b

− −

−

 − 

+ ≤ + + _ + _

 



 for 0≤ ≤ 1 with

(

)

1

1 n

n

x n

n t y

( )

n y n

b= ∆C t e +C ∆t , and = ∆t_n, we deduce,

(

)

(

)

(

(

)

)

(

( )

)

( )

(

)

(

)

( )

( )

1 3

1 1

1

1 1 2

1

2 1

1 1 2 1 1

1 1 ,

n n

n n

p

p p

p

n x n p n

y n t y n n y n p

n

p p

p x n p n

n t y n y n p

n

e C t e t C t e C t

t C

C t e C t e t

t

−

+ −

−

+ +

−

 ₋ 

   

≤ + ∆ _{ } + ∆ − + ∆ + ∆ +

 _∆ 

 

 

   _{ } 

≤ + ∆ _{ }+ ∆ _ + ∆ _{ } +



  _{ ∆ }



 



(38)

which by the inequality

(

a+b

)

p≤2p−1

(

ap+bp

)

gives

(

)

1

( )

2 1

1 n .

n

p p p p

n x n n

y n t y n y n

e ≤ + ∆C t _e+ _+ ∆C t _e _+C ∆t +

   



 (39)

Taking the mathematical expectation on both sides of (35), for sufficiently small ∆t_n we have

(

)

(

)

(

)

1

2 1

2 1 1 0

1

1 1

1 1

,

1 1 1

p n

x n

p

t y

p n

n y

N n _{n p} p N n i

x N

t_n y

i

Ch e

Ch e

Ch Ch

Ch Ch Ch

e

Ch Ch Ch

+

+

− + − −

=

 

+ _{ }

 

≤ +

− −

+ +

  _{ }  

≤_{   }+ _{ }

 

− − −

 

∑

 



 



(40)

for n=N− ⋅⋅⋅1, , 0. The terminal condition N p N p

( )

2p

y T n

e y y C t

 ₌  ₋ _{≤ ∆}

   

   

  , the time step constraint (28) and the inequality

(

0

)

2 1 1 2

2

0

1

1 ,

1 1 2

i

p N n p

c CT

i

Ch Ch h

e

Ch Ch

+ − −

= +

 _{ ≤ −}

 

−

∑

 −  (41)

lead to _en_y p ≤_ Ch2p

 

 for n=N− ⋅⋅⋅1, , 0. The proof is completed. Then we turn to estimating the error n,n

n t x

n n

z t

e =z −z .

Theorem 2. Let

(

t xn,n, t xn, n, t xn,n, t xn,n

)

t t t t

x y z Γ be the solution of the FBSDE(1) and

(

xn+1,y zn, n,Γn

)

be the solution of Scheme 1. Assume _e_yN p_= _y_T−yN p_≤C

( )

∆t_n 2p

   

  . Then for sufficiently small time step ∆t_n, we have

( )

, 0max

n n n

p

p

t x n

t n

n N z z C t

≤ ≤

 ₋ _{≤ ∆}

 

 



for 1≤ ≤ ∞p ,where C is a constant depending on T. Proof. Let n,n

n t x

n n

z t

e =z −z . From (19) and (25), we get

1

2 1

1

. n

n n

n x n n

n z t y t zj

j

t e e+ W ₊ R

=

 

∆ =_{ } ∆ _+

∑

By Lemma 2, the inequality

(

a+b

)

p≤2p−1

(

ap+bp

)

and Hölder’s inequality, we deduce

(

)

( ) ( )

1

2

1 1 1

1

2 2 2

1

2

2 2

1 2 2

2 2

, n

n n

n n

n n

p

p p

n p x n p n

n z t y t zj

j p

p

p

x n

p t y n n

p _p

x n p

p t y

t e e W R

C e t t

C e h h

+

− + −

= +

+

 

∆ ≤ _ ∆ _ +

 

 

 

≤ _{ } ∆ + ∆

 

 

 

 

 

 

≤ _{ } +

 

 

 

∑







(42)

expectation on both sides of the Equation (38) gives

( )

₁2 _{2 2}

2 _,

p _p p

p n n p

n z p y

t e C e+ h h

 

  _   _

∆ _{ }≤ _{ } +

  _   _

  (43)

by using Theorem 1 and constraint (28), leads to n p p z

e Ch

 _{ ≤}

 

 

 for n=N− ⋅⋅⋅1, ,1, 0. The proof is completed.

At last, we estimate the error n, n n t x

n n

t

e_Γ = Γ − Γ .

Theorem 3. Let

(

n,n, n, n, n,n, n,n

)

n n n

t x t x t x t x

t t t t

x y z Γ be the solution of the FBSDE(1) and

(

xn+1,y zn, n,Γn

)

be the solution of Scheme 1. Assume N p N p

( )

2p

y T n

e y y C t

 ₌  ₋ _{≤ ∆}

   

   

  . Then for sufficiently small time step ∆t_n, we have

( )

, 0

max n n

n

p

p

t x n

t n

n N C t

≤ ≤

_{Γ − Γ} _{≤ ∆}

 

 

 (44)

for 1≤ ≤ ∞p , where C is a constant depending on T. Proof. Let n,n

n t x

n n

t

e_Γ = Γ − Γ . From (26) and (31),we get

1

2 1

1

. n

n n

n x n n

n t y t j

j

t eΓ e+ N ₊ RΓ

=

 

∆ =_{ } ∆ _+

∑

₍₄₅₎

By Lemma 2, the inequality

(

a+b

)

p≤2p−1

(

ap+bp

)

and Hölder’s inequality, we deduce

(

)

( ) ( )

1

1 1 1

2 2 2

1

2

2 ₂

1 2 2

2 2

, n

n n

n n

n n

p

p p

n p x n p n

n t y t

p p

p

x n

p t y n n

p _p

x n p

p t y

t e e N R

C e t t

C e h h

+

− + −

Γ Γ

+

+

 

∆ ≤ _ ∆ _ +

 

 

 

≤ _{ } ∆ + ∆

 

 

 

 

 

 

≤ _{ } +

 

 

 









(46)

where C_p is a positive number which depends on p and the constant C in Lemma 2. Taking the mathematical expectation on both sides of the Equation (42) gives

( )

₁2 _{2 2}

2 _,

p _p p

p n n p

n p y

t e_Γ C e+ h h

 

  _   _

∆ _{ }≤ _{ } +

  _   _

  (47)

by using Theorem 1 and constraint (28), leads to _e_Γn p ≤_ Chp

 

 for n=N− ⋅⋅⋅1, ,1, 0. The proof is completed.

References

[1] Pardoux, E. and Peng, S. (1990) Adapted Solution of a Backward Stochastic Differntial Equation. Systems & Control Letters, 14, 55-61. http://dx.doi.org/10.1016/0167-6911(90)90082-6

[2] Li, Y. and Zhao, W. (2010) Lp-Error Estimates for Numerical Schemes for Solving Certain Kinds of Backward Sto-chastic Differential Equations. Statistics and Probability Letters, 80, 21-22, 1612-1617.

[3] Zhao, W., Chen, L. and Peng, S. (2006) A New Kind of Accurate Numerical Method for Backward Stochastic Diffe-rential Equations. SIAM Journal on Scientific Computing, 28, 1563-1581. http://dx.doi.org/10.1137/05063341X

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