A NOTE ON THE ALMOST SURE EXPONENTIAL STABILITY OF THE MILSTEIN METHOD FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH JUMPS

In this paper we consider the Milstein method to investigate the almost sure exponential stability for nonlinear stochastic delay differential equations (SDDEs) with jump diffusion, that is, processes that change only by jumps. The class of jump-diffusion SDDEs that admits explicit solutions is rather limited. In this regard, we employ the continuous and discrete semimartingale convergence theorem. It is shown that the Milstein method reproduces the almost sure exponential stability under an additional linear growth condition on drift and the function Lg. In the numerical section we present the results by simulating the method on kind of equations. AMS Subject Classification: 65C20, 60H35, 65C30


Introduction
Jump diffusion is a stochastic process that involves jumps and diffusion.Mod-els that incorporate jumps have become increasingly popular in finance and several areas of science and engineering.They have important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, pattern theory and computational vision (see [27]).In particular, they are used in mathematical finance in order to simulate asset prices, interest rates and volatilities (see [2] and [7]).Jump models also are employed, appear in many other application areas and have proved successful at describing unexpected, abrupt changes of state [4].Since only a limited class of jump-diffusion problems admit explicit solutions, it is imperative to develop numerical methods to solve these problems.
In special cases, stochastic delay differential equations (SDDEs) are a type of SDEs, which has been discussed in variety of sciences (see [9], [12] and [23]).Since most of these equations cannot be solved explicitly, numerical approximations become an important tool in studying the properties of these stochastic systems (see [1], [3], [10], [17], [20] and [26]).Since stability is one of the major concerns in systems analysis, the stability of numerical methods becomes also one of the main tools to examine the stability of the exact solution of the stochastic systems (see [18], [21] and [24]).Using the technique based on the continuous semimartingale convergence theorem, Mao [16] investigated the almost sure asymptotic stability of SDDEs.Now efforts are being made to bring SDDEs with jumps up to a similar level.For SDDEs with jumps, the existing literature concerns mainly stability analysis of numerical methods.For example, Wang et al [11] constructed the semiimplicit Euler method for stochastic differential delay equation with jumps.Tan and Wang [5] investigated the mean-square stability of the explicit Euler method for linear SDDEs with jumps.Zhang et al [8] derived some criteria on pth moment stability and almost sure stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching.Li et al [19] constructed compensated stochastic theta methods (CSTM) for stochastic differential delay equations with Poisson jumps, and got the mean square P-stability by some restriction on time stepsize.Also, the stability of Milstein method has been applied to SDDEs.For example, [25] discusses the mean square stability of SDDEs with constant coefficients and [22] obtains also mean-square stability for SDDEs with integral part.Based on these papers, to the best of our knowledge, the alsmost sure exponential stability of Milstein method for SDDEs with jump has never been considered so far.In this paper, we derive a sufficient condition of the almost sure exponential stability of the Milstein method for SDDEs with jumps.These stability results show that the Milstein method reproduces the almost sure ex-ponential stability under an additional linear growth conditions on drift and the function L 1 g.
The rest of the paper is organized as follows.Section 2 begins with notations and preliminaries, and establish the continuous semimartingale convergence theorem and discrete semimartingale convergence theorem.In Section 3 the Milstein method is introduced for SDDEs with jump.Section 4 examines the conditions under which the Milstein method can reproduce the almost sure exponential stability.Section 5 gives the numerical experiments to confirm the theoretical results.

Preliminaries and notations
Throughout this paper, unless otherwise specified, we use the following notations.Let | • | denotes both the Euclidean norm in R n and the trace (or Frobenius) norm in R n×d .The inner product of x, y ∈ R n is denoted by x, y or x T y.If A is a vector or matrix, its transpose is denoted by A T .If A is a matrix, its trace norm is denoted by |A| = trace(A T A). a ∨ b represents max{a, b} and a ∧ b denotes min{a, b}.Let (Ω, F, P) be a complete probability space with a filtration {F t } t≥0 , which is right continuous and satisfies that each F 0 contains all P-null sets.Denote by C([−τ, 0]; R n ) the family of continuous functions from [−τ, 0] to R n equipped with the norm ψ := sup −τ ≤t≤0 |ψ(t)|.Let W (t) be a d-dimensional standard Brownian motion defined on this probability space and N (t) be a scalar Poisson process with parameter λ > 0 and independent of the Brownian motion.Let f : R Let us consider the n-dimensional SDDE with jump-diffusion of the form where the delay τ > 0 is a fixed constant, x(t − ) denotes lim s→t − x(s).As a standing hypothesis, we assume that the functions f , g and h satisfy the local Lipschitz condition: Assumption 1.The functions f , g and h satisfy the local Lipschitz condition, that is, for each j > 0 there exists a positive constant C j such that for all t ≥ 0 and any x, y, x, y ( To be precise, we give the definition on the almost sure exponential stability of the SDDEs with jumps.Definition 1.The solution x(t) to Equation ( 1) is said to be almost surely exponentially stable if there exists a constant η > 0 such that lim sup for any initial data In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem, gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process starting at zero.The following lemmas will play important roles in this paper.
Lemma 3. (Discrete Semimartingale Convergence Theorem, see [6] and [29]).Let {A i }, {U i } be two sequences of nonnegative random variables such that both A i and U i are F i -measurable for i = 1, 2, • • •, and A 0 = U 0 = 0 a.s.Let M i be a real-valued local martingale with M 0 = 0 a.s.Let ζ be a nonnegative F 0 -measurable random variable.Assume that {X i } is a nonnegative semimartingale with the Doob-Mayer decomposition that is, both X i and U i converge to finite random variables.
We are now ready to formulate our first result, which provide us with a criterion on the almost sure exponential stability of the exact solution to Equation (1).Theorem 4. Let Assumption 1 holds.Assume that there exist positive constants α i , β i , γ i , i = 1, 2 such that for any x, y ∈ R n and t ≥ 0 then in [19] Li et al, show that for any given initial data ψ(t), there exists a unique global solution x(t) to Equation (1) and the solution has the property that lim sup where γ > 0 is the unique positive root of

The Milstein method on SDDEs with jumps
Now we introduce the Milstein approximation {y k } k≥0 as follows: where L 1 = g(x) ∂ ∂x , and stepsize ∆t = τ m for a integer m, y k = ψ(k∆t) for k = −m, . . ., 0, y 0 = ψ(0), whose value of ∆t may change with each appearance.The Brownian increments is defined as ∆W k := W (t k+1 ) − W (t k ) and ∆N k := N (t k+1 ) − N (t k ) represent the Poisson process, respectively.Assumption 2. Assume that the function L 1 g satisfy the local Lipschitz condition, that is, there exists a constant K 1 such that for any x, y, x, y ∈ R n and t > 0, For the purpose of stability, we assume that f (t, 0, 0) = g(t, 0, 0) = h(t, 0, 0) = 0.This shows that Equation (1) admits a trivial solution, then inequality (16) becomes To be precise, the definition on the almost sure exponential stability of the numerical solutions is given as follows: Definition 5.The numerical solution {y k } k≥0 to Equation ( 15) is said to be almost surely exponentially stable, if there exists a constant η > 0 such that lim sup for any initial data ψ(k∆t) for k = −m, −m + 1, . . ., 0.

Stability analysis of the Milstein method on SDDEs with jumps
In this section, we assume that f , g and h satisfy the local lipschitz condition.Now we investigate the almost sure stability analysis of Milstein approximation {y k } k≥0 with jumps.
Theorem 6.Let conditions (8)-( 12) hold and the function L 1 g satisfies the linear growth condition (17).If there exists a positive constant K 2 such that f satisfies the linear growth condition then there is a stepsize bound ∆t * > 0, such that for any ∆t < ∆t * , the Milstein approximation {y k } k≥0 has the property where η > 0 is the unique root of the equation and ε ∈ (0, η 2 ) is arbitrary.

Numerical experiments
In this section, we introduce a model of SDDE with jump process whose the coefficients are nonlinear and its simulation to illustrate the almost sure exponential stability of the Milstein method.

Conclusion
In this paper, we consider the almost sure exponential stability of the Milstein method for the SDDEs with Poisson jumps.For this purpose, we derive some sufficient conditions for the drift, diffusion and Poisson coefficients.Also, the semimartingale theorem of Doob is given, and using martingale property of the compensated Poisson process, we reach the almost sure exponential stability.
Finally, we present an example simulated and the numerical experiments verify the results obtained from theory.

Figure 1 :
Figure 1: Almost sure exponential stability of y n with initial data x(t) = 2 4 .