2018 International Conference on Physics, Computing and Mathematical Modeling (PCMM 2018) ISBN: 978-1-60595-549-0
Stability of Exponential Euler Method for Linear Stochastic Delay
Differential Equations
Ling ZHANG
*and Guo-qing LIU
Mathematical Department of Teacher Education Institute, Daqing Normal University, Daqing, 163712, P.R. China
*Corresponding author
Keywords: Stochastic delay differential equation, Exponential euler method, Lipschitz condition, Itˆo
formula, Strong convergence.
Abstract. The main purpose of this paper is to investigate the exponential stability in mean square of the exponential Euler method to linear stochastic delay differential equations (LSDDEs). The classical stability theorem to LSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to LSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to LSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to LSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.
Introduction
Stochastic modeling has come to play an important role in many branches of science and industry. Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics and finance. Most stochastic differential equations (SDEs) are nonlinear and cannot be solved explicitly, whence numerical solutions are required in practice. Numerical solutions to SDEs have been discussed under the Lipschitz condition and the linear growth condition by many authors (see [1],[2],[3],[4],). Many authors have discussed numerical solutions to stochastic delay differential equations (SDDES) (see [5], [6] ,[7]). The stability of the implicit Euler scheme to SDEs is known for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to LSDDEs is proved to share the stability for any step size by the property of logarithmic norm.
Preliminary Notation and the Exponential Euler Method
Let ( ) ( 1( ), , ( ))
T d
B t B t B t be a d-dimensional Brownian motion defined on the probability space
( , , ) F P . Throughout this paper, we consider the following linear stochastic delay differential
equations:
( ) ( ( ) ( )) ( ( ) ( )) ( ), [0, ]
( ) ( ), [ , 0],
dx t Ax t Bx t dt Cx t Dx t dB t t T
x t t t
(1)
where 0, 0,{ ( ), [ , 0]; },
n n n
T t t R AR the matrix [8], B C D, , are constants. By the definition of stochastic differential, this equation is equivalent to the following stochastic integral equation:
( ) ( )
0 0
( ) At t A t s ( ( ) ( )) t A t s ( ( ) ( )) ( ), 0.
x t e
e Ax t Bx t ds
e Cx t Dx t dB s tLet
h m
be a given step size with integer m1; and let the grid points tn be defined by
( 0,1, 2, )
n
t nh n
. We consider the exponential Euler method to (1)
1 ( ) ( ) ,
Ah Ah Ah
n n n m n n m n
y e e Ay By h e Cy Dy B (3)
where Bn B t( )n B t( n1),n0,1, 2, ,yn is approximation to the exact solution x t( )n . The continuous exponential Euler method approximate solution is defined by
( ) ( )
0 0
( ) At t A t s ( ( ) ( )) t A t s ( ( ) ( )) ( ), 0.
y t e
e Az t Bz t ds
e Cz t Dz t dB s t(4)
where z t( ) k 0y Ik [kh k,( 1) )h ( )t
with IA denoting the indicator function for the set A. It is not
difficult to see that y t
n z t
n yn for n0,1, 2, . That is, the step function ( )z t and the continuous exponential Euler solution ( )y t coincide with the discrete solution at the grid point.Exponential Stability in Mean Square
In this section, we give the exponential stability in mean square of the exact solution and the exponential Euler method to semi-linear stochastic delay differential equations (1).
Stability of the Exact Solution
In this subsection, we will show the exponential stability in mean square of the exact solution to semi-linear stochastic delay differential equations (1). Next we will give the main content of this subsection.
Theorem 3.1. If 1 2 [ ] 4 µ A
B2C2D2
0, then the solution of equations (1) with the initial data0([ , 0], )
b n
F
C R
is exponentially stable in mean square, that is,
1 2
2 1 2 ln( ( ))
| ( ) | ( ) | | t B , 0,
E x t B E e t
Where 1 2 1 2 2 2 2 2 2
1 2
1
( ) B B (1 B ), 1 2 [ ] 2( ), 2( ).
B e e B A B C D B B C D
B
By Ito’s formula and the delay term of the equation, we give the proof of Theorem 3.1. The highlight of the proof is that we give the mean square boundedness of the solution to the equation by
dividing the interval into
0,
, , 2
, , [k,
k1
]. Then we give a proof of the conclusion by t0, t2 , t4 , , t2n. In the process of dealing with the semi-linear matrix, we use the definition of the matrix norm.Definition 3.1.[7] SDDEs (1) are said to be exponentially stable in mean square if there is a pair of positive constants λ and µ such that for any initial data
0([ , 0], )
b n
F
C R
2 2
| ( ) | | | t, 0.
E x t E e t
We refer to λ as the rate constant and to µ as the growth constant.
Definition 3.2.[9] The logarithmic norm µ[A] of A is defined by
0
1
[ ]A lim I A
2 0
,
[ ] max .
|| ||
A A
(5)
Lemma 3.1. Let 1 2 1
1
( ) B t B (1 B t).
B t e e
B
If B10,B2 0 and B1B20 then for all
0, 0 ( ) 1
t B t and B t( )is decreasing.
Proof . It is known from B10,B2 0 and B1B2 0 then for all t0,
1
1 2 2
1 1
( ) B B B t B 0
B t e
B B
and
1
1 2 1 1 2
1 1
( )( 1)
( ) 1 ( 1) 1 1 1.
B t
B t B B t B B e
B t e e
B B
For all t0 , we compute 1
'
1 2
( ) ( ) B t 0.
B t B B e Thus B t( ) is decreasing. The proof is
complete.
Proof of Theorem 3.1. By Ito’s formula and Definition 3.2, for all t0; we have
2 2
2 2
1 2
| ( ) | [ 2 ( ), ( ) ( ) | ( ) ( ) | ]
2 ( )( ( ) ( )) ( )
[ | ( ) | | ( ) | ] 2 ( )( ( ) ( )) ( ).
T
T
d x t x t Ax t Bx t Cx t Dx t dt
x t Cx t Dx t dB t
B x t B x t dt x t Cx t Dx t dB t
(6)
Where
2 2 2 2 2 2
1 1 2 [ ] 2( ), 2 2( )
B A B C D B B C D . Let V x t( , )eB t1 | ( ) | ,x t 2
by Ito’s formula, we obtain
1 1 1 1
1
2 2 2 2
1 2
| ( ) | | ( ) | | ( ) | | ( ) |
2 ( )( ( ) ( )) ( )
B t B t B t B t
B t T
d e x t B e x t dt e d x t B e x t dt
e x t Cx t Dx t dB t
(7)
Integrating (7) from 0 to t and taking expected values gives
1 2 2 1 2
2 0
| ( ) | | (0) | t | ( ) | .
B t B s
e E x t E x B
e E x s dsFor any t[0,], we have
1 1
2 2 2 2
1
| ( ) | [ B t B (1 B t)] | | ( ) | | .
E x t e e E B t E
B
For any t[ ,2 ], we have
1( ) 1( )
2 2 2 2
1
| ( ) | [ B t B (1 B t )] | | ( ) | | .
E x t e e E B t E
B
Repeating this procedure, for all t[k, (k1)],we can show
2 2
| ( ) | ( ) | | .
E x t B tk E
So, for anyt0; we have
2 2
| ( ) | | | .
1 1
2 2 2 2
1
| ( ) | [ B t B (1 B t)] | | ( ) | | .
E x t e e E B t E
B
Especially, we can see
2 2
| (2 ) | (2 ) | | .
E x B E
For any t2 , we have
2 2 2 2
| (4 ) | ( ) (2 ) | | ( ) | | .
E x B B E B E
For any t4 , we can see that
1( 4 ) 1( 4 )
2 2 2 2 2 2
1
| ( ) | ( )[ B t B (1 B t )] | | ( ) ( 4 ) | | .
E x t B e e E B B t E
B
For any t0 there is an integer n such that t2n , repeating this procedure, we can show
2 2 2
| ( ) | n( ) ( ) | | n( ) | | .
E x t B B tn E B E (8)
By (8) and Lemma 3.1, we obtain
1 2
1 1
2 2
1 1
2 2
1 2
2 2
2 ln( ( ) ) 2
(2 ) ln( ( ) ) 2 ln( ( ) )
2 ln( ( ) ) 2 ln( ( ) )
1 2 ln ( )
| ( ) | ( ) | |
| |
| |
| |
( ) | | ,
n
n B
n t B t B
B t B
t B
E x t B E
e E
e E e
e E e
B E e
which proves the theorem.
Stability of the Exponential Euler Method
In this subsection, under the same conditions as those in Theorem 3.1, we will obtain the exponential stability in mean square of the exponential Euler method (4) to LSDDEs (1) in Theorem 3.2.
Definition 3.3.[7] Given a step size h m
for some positive integer m, the discrete exponential
Euler method is said to be exponentially stable in mean square on SDDEs (1) if there is a pair of positive constants and such that for any initial data
0([ , 0], )
b n
F
C R
,
2 2
| n| | | nh, 0.
E y E e n
Theorem 3.2. If 1 2 [ ] 4 µ A
B2C2D2
0 then for all h0 the numerical method to equations (1) is exponentially stable in mean square, that is1 2 1 2
ln( )
2 1 2
1 2 0
| n| ( ) | | nh A A ,
E y A A E y e
Where
2 [ ] 2 2 2 2 2 2 2
1 (1 ) 2( ) ),
A h
A e B C D h B C D h h
2 [ ] 2 2 2 2 2 2 2
2 ( ) 2( ) ).
A h
A e B C D h B C D h
2
2 [ ] 2 2 [ ] 2 2 2 [ ] 22 [ ] 2
| | | | | | | ) | | | )
2 ( , | ) ,
| A h A h A h
n nh n n n m nh n n m nh
A h
n n n m nh
E y F e E y e E Ay By F h e E Cy Dy F h
e E y Ay By F h
Taking expectations on both sides, we obtain that
2 2 2
1 2
| n| | n| | n m| .
E y A E y A E y
Where
2 [ ] 2 2 2 2 2 2 2
1 (1 ) 2( ) ),
A h
A e B C D h B C D h h
2 [ ] 2 2 2 2 2 2 2
2 ( ) 2( ) ).
A h
A e B C D h B C D h for all h0; which implies
2
2 2 2 2 2 2 2 ( 2 [ ] ) 2 [ ]
1 4( ) 2( ) 1 2 [ ] .
2!
A h
A h
h B C D h B C D h A h e
That is A1A2 1 for all h0. for all n 1, 2, , we have
1 2
1 1
1
( 1) ( 1)
( 1)
1 2 1 2
1 2 1
2 1 2
[ ] 1 ([ ] 1) ln( )
2 2 2 1 2 1 2
1 2 1 2 0 0
[ ]( 1) ln( ) { }( 1) ln( ) ln( )
2 2
1 1
0 0
ln( )
1 2
1 2 0
| | | | | | ( ) | | | |
| | | |
( ) | | .
m h m h
m h
n n
A A
m m
n n n m
n n
m h A A m h A A nh A A
m m
nh A A
E y A E y A E y A A E y e E y
e E y e E y e
A A E y e
The proof is completed.
Numerical Experiments
In this section, we give several numerical experiments in order to demonstrate the results about the strong convergence and the exponential stability in mean square of the numerical solution for equations(1). We consider the test equation
1 2 1 2
( ) [ ( ) ( )] [ ( ) ( )] ( ) 0
dx t a x t a x t dt b x t b x t dB t t (9)
Example 9. When a1 5,a2 1, b12,b2 0.5, 1 t, 1. We can show the stability of the exponential Euler method to (3). In Figure 1, all the curves decay toward to zero when
1 , 1 , 1 , 1 .
2 8 32 128
h h h h So we can consider that our experiments are consistent with our
Acknowledgement
This work is supported by the Youth Science Foundations of Heilongjiang Province of P.R. China (No.QC2016001)
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