Numerical solutions of hybrid stochastic differential delay equations under the generalized khasmins

Mathematical Computation December 2014, Volume 3, Issue 4, PP.112-121

Numerical Solutions of Hybrid Stochastic Differential Delay Equations under the Generalized Khasminskii-Type Conditions Liangjian Hu #, Yanke Ren Department of Applied Mathematics, Donghua Univerisity, Shanghai 201620, China #Email: ljhu@dhu.edu.cn

Abstract The main aim of this paper is to discuss the numerical solution of nonlinear hybrid stochastic differential delay equations (SDDEs), where the linear growth condition is replaced by the more general Khasminskii-type condition. This generalized Khasminskii condition covers a wide class of highly nonlinear hybrid SDDEs whose solution is not explicit. We establish the existence-and-uniqueness theorem and prove the Euler-Maruyama approximations converge to the true solution in probability.

Keywords: Khasminskii-Test; Euler-Maruyama Approximations; Hybrid SDDE; Convergence in Probability

1 INTRODUCTION The classical existence and uniqueness theorem for hybrid stochastic differential delay equations (SDDEs) requires the coefficients of the underlying SDDEs satisfying the local Lipschitz condition and the linear growth condition [1, 2, 5, 8] . But there are lots of SDDEs which do not satisfy the linear growth condition[3,4,6,9,11,12].

In 2005, Mao and Rassias [7] established a generalized Khasminskii-type theorem which covers a wide class of nonlinear SDDEs. In 2011, Mao [10] proved the convergence of Euler-Maruyama (EM) numerical solutions of SDDEs under the generalized Khasminskii-type conditions. In this paper, we will prove the EM approximations of hybrid SDDEs under the generalized Khasminskii-type conditions converge to the true solution in probability.

The paper is organized as follows: we first introduce necessary notations and establish the generalized Khasminskiitype existence and uniqueness theorem for hybrid SDDEs in Section 2. Then we establish the EM approximations of the hybrid SDDEs under the Khasminskii-type conditions in Section 3 and prove their convergence in probability in Section 4. Finally, we give an example in Section 5 to show that the EM numerical method can be applied to approximate the true solution of highly nonlinear SDDEs.

2 THE KHASMINSKII-TYPE THEOREM FOR HYBRID SDDES Throughout this paper, unless otherwise specified, we use the following notation. Let be the Euclidean norm in . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by . Let and . Denote by the family of continuous functions from to with the norm . Let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is increasing and right continuous while contains all ‐ null sets). Let be an ‐ dimensional Brownian motion defined on the probability space. Let , be a right‐continuous Markov chain on the probability space taking values in a finite state space with generator given by

where

. Here

is the transition rate from to

if

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while

We assume that the Markov chain

is independent of the Brownian motion

.

Consider a nonlinear hybrid SDDE (2.1) on

, where

is a constant and

The initial data are given by (2.2) Assumption 2.1 (The local Lipschitz condition). For each integer

for those

with

there is a positive constant

and any

such that

.

Let denote the family of all continuous functions from family of all continuous non�negative functions defined on continuously twice differentiable in . Given by

to

. Denote by such that for each , we define

the , they are the function

where

Let us emphasize that

is defined on

while

Assumption 2.2 There are two functions constants with , such that

on and

. , as well as three nonnegative

(2.3) and (2.4) for all

.

Theorem 2.3 Let Assumptions 2.1 and 2.2 hold. Then for any given initial data (2.2),there is a unique global solution to the hybrid SDDE (2.1) on . Moreover, the solution has the properties that (2.5) Proof. Since the coefficients of the hybrid SDDE (2.1) are locally Lipschitz continuous, for any given initial data (2.2) there is a unique maximal local solution on , where is the explosion time ([8, Theorem 7.12 on page 278]). Let be an integer sufficiently large for . For each integer , define the stopping time

where throughout the paper, we set Set , whence

(as usual denotes the empty set). Clearly, a.s.. If we can show that a.s., then - 113 www.ivypub.org/mc

is increasing as a.s., and

. is the

unique global solution to Eq(2.1) on

.

By the generalized It么 formula ( [8, Lemma 1.9 on page 49]) and condition (2.4), we can show that, for any and ,

(2.6) But

Substituting this into (2.6) yields

(2.7) where

. Recalling that

, we obtain

Define

Then Hence But by condition(2.3), , namely Since that

as

.We can therefore let

is arbitrary, we must have that

Letting

in the inequality above to obtain the

as required. To show assertion (2.5), we see from (2.7)

, we then have

as required.

Corollary 2.4 Let Assumptions 2.1 and 2.2 hold. Let large integer , dependent on and , such that

where

and

be arbitrary. Then there is a sufficiently

is the same as defined in Theorem 2.3.

Proof. See from the proof of theorem 2.3.

3 THE EULER-MARUYAMA APPROXIMATIONS AND THEIR PROPERTIES Let

be

a

positive

integer

and

be

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the

step

size.

Define

for

. To define the Euler-Maruyama approximate solution, let us recall the property of the embedded discrete-time Markov chain: let . Then ( [8, 2nd paragraph on page 112]) is a discrete-time Markov chain with the one-step transition probability matrix

Set

for

, and (3.1)

Where

.

Let

(3.2) Define

for

, while for

Both and are the continuous-time approximations, useful to know that for .

Lemma 3.1 Let Assumption 2.1 holds. Let

is computable while, in general,

(3.3) is not. It is

be an arbitrary number and let be sufficiently large integer for

Define the stopping time Let

be any integer sufficiently large for (3.4)

Then (3.5) where

Proof. For

, let

Hence, by Assumption 2.1, for

be the integer part of

. Then

, such that

Then

In the same way,

and - 115 www.ivypub.org/mc

and

So we get

Therefore, we have( [10, Lemma 3.1])

as required.

Lemma 3.2 Let Assumptions 2.1 and 2.2 hold. Then for any pair of large and sufficiently small such that where

and

, there is a sufficiently

is as defined in Lemma 3.1.

Proof. The proof of this lemma is very technical so we divide it into 4 steps.

Step 1: By the generalized It么formula, we have that, for is that

, the coefficients of the term of

of

(3.6) and

Let

is defined by

be

any sufficiently large integer. , by Assumption 2.1, we compute

For

with

where thought this proof denotes a positive constant, independent of (but dependent on etc. of course), which may change from line to line. Substituting this into (3.6) we obtain that, for ,

(3.7) Hence, for

,

(3.8) But

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for

([10, Lemma 3.2]). Substituting this into (3.8) yields that, for

,

(3.9) where

.

Step 2: Let us now consider

. By Assumption 2.2, we derive from (3.9) that

(3.10) Hence, (3.11) which is determined by the initial data. Having this, we further derive from

where (3.10) that

Hence

(3.12) . If follows from (3.9) that

Step 3: Let us further consider

(3.13) where

Hence (3.14) and

(3.15) Furthermore, let us estimate

, clearly

It is easy to verify We hence compute, using (3.12), that - 117 www.ivypub.org/mc

Hence,

Step 4: Repeating this procedure as in Steps 2 and 3 we can show that

where

and

(3.16) are two constants dependent on as well as

Now, for any

and then choose

but independent of . Recalling the definition , we then have

(3.17) sufficiently large for

, choose

sufficiently small for

It then follows from (3.17) that as required.

4 CONVERGENCE IN PROBABILITY To show the results, both continuous-time approximations, probability. Let be an integer sufficiently large for

for

and

, will converge to the true solution . Define

in

, where of course set

By Assumption 2.1, both

and

are globally Lipschitz continuous. Hence, given the initial data (4.1)

the hybrid SDDE (4.2) has a unique global solution

on

( [8, Theorem 7.10 on page 277]). Moreover (4.3)

for all

, where

is the same as defined in Theorem 2.3.

Applying the EM method to the hybrid SDDE (4.2) we can define the corresponding continuous-time approximations, and , as we did in the beginning of Section 3. It is known that ( [8, Theorem 7.29 on page 296])

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(4.4) However, it is straightforward to see that

for all , where following lemma.

(4.5) is the same as defined in Lemma 3.1. Combining these results we immediately obtain the

Lemma 4.1 For all sufficiently large integer , we have (4.6) We can now show our first convergence result.

Theorem 4.2 For any

, (4.7)

Proof. Let

be arbitrarily small. Set

By Theorem 2.4 and Lemma 3.2 there is a pair of

Let

and

such that

and compute

But

By Lemma 4.1, we see that for all sufficiently small

Consequently, for all sufficiently small

,

,

as required.

Theorem 4.3 For any

,

(4.8) Proof. By Lemma 3.1 and Lemma 4.1, in the same way as Theorem 4.2 was proved, it’s easy to get the assertion.

5 EXAMPLE In this section we will discuss an example to show that the EM numerical method can be applied to approximate the solutions of highly nonlinear SDDEs. Example 5.1. Consider a one-dimensional hybrid SDDE

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with the initial data , where is a right-continuous Markov chain on the state space

(1.1) is a one-dimensional Brownian motion and with the generator

while

Since and don’t satisfy the linear growth condition, classical theory in [1‐3] cannot applied to the example. Obviously the coefficients of this hybrid SDDE obey the local Lipschitz condition described in Assumption 2.1. Hence, to show that the EM numerical method can be applied to approximate the true solution, we only need verify Assumption 2.2. Let us define by

for

. Then the operator

has the form

and

If we define

for

, then

for , where and . So Assumption 2.2 is satisfied and by Theorem 4.2 and Theorem 4.3 we know that the EM numerical method can be applied to approximate the solution of hybrid SDDE (5.1).

ACKNOWLEDGMENT The authors would like to thank Natural Science Foundation of China (grant 11071037) for its financial support.

REFERENCES [1]

D.J. Higham, X. Mao and A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis 40(3) (2002): 1041--1063.

[2]

D.J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM Journal on Numerical Analysis 45 (2) (2007): 592—609.

[3]

L. Hu, X. Mao, and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Systems Control Letters 62(2013): 178-187.

[4]

M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 467 (2130) (2011): 1563-1576.

[5]

P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992

[6]

X. Mao, Stochastic versions of the LaSalle theorem, Journal of Differential Equations 153 (1) (1999): 175-195.

[7]

X. Mao and M.J.Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl. 23 (2005): 1045-1069.

[8]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.

[9]

X. Mao, C. Yuan and G. Yin, Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching under non-Lipschitz conditions, Journal of Computational and Applied Mathematics 205(2007): 936-948.

[10] X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Applied Mathematics and Computation 217 (2011): 5512-5524.

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[11] X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Computational and Applied Mathematics 238 (2013): 14-28.

[12] M. Song, L. Hu, X. Mao and L. Zhang, Khasminskii-Type theorems for stochastic functional differential equations, Discrete and Continuous Dynamical Systems - Series S, 18(6) (2013): 1697-1714.

AUTHORS 1Liangjian

Department

Hu is a Full Professor with the

of Strathclyde, Glasgow, Scotland, UK. His research interests

Donghua University, Shanghai, China. He

include stochastic control, fuzzy system, and stochastic

received the B.S. degree from Anhui

differential equation with applications. Dr.

Normal University, Wuhu, Anhui, China,

Hu is the author of 4 books and more than

the M. S. degree and the Ph. D degree

50 research papers.

Donghua

Applied

and in the Department of Mathematics and Statistics, University

Mathematics,

from

of

University,

Shanghai,

China, in 1985, 1988, and 2003, respectively. He had been a postdoctoral researcher in the Department of Electrical Engineering, National Tsinghua University, Hsin-Chu, Taiwan,

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2Yanke

Ren is currently a postgraduate

student at the Department of Applied Mathematics, Shanghai, China.

Donghua

University,