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.
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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. - 120 www.ivypub.org/mc
[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,