Fuke Wu

Choice of θ and mean-square exponential stability in the stochastic theta method of stochastic differential equations

This paper examines the relationship of choice of ? and mean-square exponential stability in the stochastic theta method (STM) of stochastic differential equations (SDEs) and mainly includes the following three results: (i) under the linear growth condition for the drift term, when ? ? 0 , 1 / 2 ) , the STM may preserve the mean-square exponential stability of the exact solution, but the counterexample shows that the STM cannot reproduce this stability without this linear growth condition; (ii) when ? ? ( 1 / 2 , 1 ) , without the linear growth condition for the drift term, the STM may reproduce the mean-square exponential stability of the exact solution, but the bound of the Lyapunov exponent cannot be preserved; (iii)?when ? = 1 (this STM is called as the backward Euler-Maruyama (BEM) method), the STM can reproduce not only the mean-square exponential stability, but also the bound of the Lyapunov exponent. This paper also gives the sufficient and necessary conditions of the mean-square exponential stability of the STM for the linear SDE when ? ? 0 , 1 / 2 ) and ? ? 1 / 2 , 1 ] , respectively, and the simulations also illustrate these theoretical results.

Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations

The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766-1793 (1996); ibid. 56, 1794-1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.

Convergence of Numerical Approximation for Jump Models Involving Delay and Mean-Reverting Square Root Process

The mean-reverting square root process with jump has been widely used as a model on the financial market. Since the diffusion coefficient in the model does not satisfy the linear growth condition and local Lipschitz condition, we can not examine its properties by traditional techniques. To overcome the difficulties, we develop several new techniques to examine the numerical method of jump models involving delay and mean-reverting square root. We show that the numerical approximate solutions converge to the true solutions. Finally, we apply the convergence to examine a path-dependent option price and a bond in the financial pricing.

Almost sure exponential stability of the backward Euler-Maruyama scheme for stochastic delay differential equations with monotone-type condition

This paper is a continuation of our previous paper, in which, the second author, with Mao and Szpruch examined the almost sure stability of the Euler-Maruyama (EM) and the backward Euler-Maruyama (BEM) methods for stochastic delay differential equations (SDDEs). In the previous results, although the drift coefficient may defy the linear growth condition, the diffusion coefficient is required to satisfy the linear growth condition. In this paper we want to further relax the condition. Under monotone-type condition, this paper will give the almost sure stability of the BEM for SDDEs whose both drift and diffusion coefficients may defy the linear condition. This improves the existing results considerably.

On a Class of Nonlocal Stochastic Functional Differential Equations with Infinite Delay

This article considers a class of nonlocal stochastic functional differential equations with infinite delay whose coefficients are dependent the pth moment and establishes the existence-and-uniqueness theorem under the conditions that are similar to the classical linear growth condition and the Lipschitz condition. Compared with the existing results, the conditions of this article are easier to test.

Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations

This paper examines convergence and stability of the two classes of theta-Milstein schemes for stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients: the split-step theta-Milstein (SSTM) scheme and the stochastic theta-Milstein (STM) scheme. For \theta\in[1/2,1], this paper concludes that the two classes of theta-Milstein schemes converge strongly to the exact solution with the order 1. For \theta \in [0,1/2], under the additional linear growth condition for the drift coefficient, these two classes of the theta-Milstein schemes are also strongly convergent with the standard order. This paper also investigates exponential mean-square stability of these two classes of the theta-Milstein schemes. For \theta\in(1/2, 1], these two theta-Milstein schemes can share the exponential mean-square stability of the exact solution. For \theta\in[0, 1/2], similar to the convergence, under the additional linear growth condition, these two theta-Milstein schemes can also reproduce the exponential mean-square stability of the exact solution.

JUMP SYSTEMS WITH THE MEAN-REVERTING γ-PROCESS AND CONVERGENCE OF THE NUMERICAL APPROXIMATION

In this paper, a class of jump systems with the mean-reverting γ-process are considered in finance. The analytical properties including the positivity, boundedness and pathwise estimations of the solution are discussed. Moreover, the authors show that the Euler–Maruyama approximate solutions converge to the true solutions in probability. Finally, the authors apply the convergence to examine a bond and a path-dependent option price in the financial pricing.

The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations

Based on the martingale theory and large deviation techniques, we investigate the p th moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the p th moment exponential stability of Euler–Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition.

Weak solution of stochastic differential equations with fractional diffusion coefficient

ABSTRACT In stochastic financial and biological models, the diffusion coefficients often involve the term , or more general |x|r, r ∈ (0, 1). These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This article establishes the existence of the weak solution for this class of stochastic differential equations by using the martingale representation and weak convergence methods.