Parametric continuity of solutions to stochastic functional differential equations with poisson perturbations

The small-parameter method and the notion of averaged system are used to analyze the asymptotic stability in the mean square of the original system of stochastic differential equations. The stability of a system with continuous perturbations is considered. It is proved that the small-parameter method can be applied to stochastic differential equations with discontinuous trajectories, i.e., that stochastic differential depends on the Poisson integral.     

随机中立型泛函微分方程的Lasalle定理

随机型的Lasalle定理是研究随机系统的稳定性的重要理论工具.本文应用It公式、半鞅收敛定理与ko lm ogorov-Cˇentsov定理等随机分析知识,以及H lder不等式等技巧,首次建立了一般随机中立型泛函微分方程的Lasalle定理,由此得到一些有用的随机稳定性判据.本文所建立的结果涵盖并推广了已有文献的结论.最后给出了一个例子说明本文结果的有效性.     

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 74–88, January–February 2008.     

Multiple stochastic integrals with Mathematica

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectation of products of multiple stochastic integrals. Well-known recursive relationships between these multiple integrals make it possible to express any product of them as a linear combination of integrals of the same type. This article describes how, exploiting the symbolic character of Mathematica, main recursive properties and rules of Itô and Stratonovich multiple integrals can be implemented. From here, a routine that calculates the expectation of any polynomial in multiple stochastic integrals is obtained. In addition, some new relations between integrals, found with the aid of the program, are shown and proved.     

Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients

Under a non-Lipschitz condition with the Lipschitz condition being considered as a special case and a weakened linear growth condition, the existence and uniqueness of mild solutions to stochastic neutral partial functional differential equations (SNPFDEs) is investigated. Some results in Govindan (2003, 2005) [2], [6] are generalized to cover a class of more general SNPFDEs.     

New criteria on asymptotic behavior of neutral stochastic functional differential equations

This paper investigates the asymptotic behavior of neutral stochastic functional differential equations (NSFDEs) under both the local Lipschitz condition and the one dependent on the diffusion operator and on a coercivity term, which is more general than the classical growth condition. Some sufficient conditions for stability with general decay rate and boundedness of NSFDEs are derived via the Lyapunov analysis method and some stochastic analysis techniques. Our results not only cover a wide class of highly nonlinear NSFDEs but they can also deal with general stability issues including the polynomial stability and the exponential stability. Finally, an illustrative example is provided to show the effectiveness of our theoretical results.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

New-type stability theorem for stochastic functional differential equations with application to SFDSs with distributed delays

In this paper, a new-type stability theorem for stochastic functional differential equations (SFDEs) is established, which is not a direct copy of the basic stability theorem for deterministic functional differential equations (DFDEs). By the new-type stability theorem, one can use the most simple Lyapunov functions and employ the equations repeatedly to deal with the delayed terms encountered conveniently and to carry out stability criteria for the equations. Based on the theorem, a practical stability theorem in accordance with the Lyapunov function method is also established, and then the asymptotic stability of SFDEs with distributed delays in the diffusive terms is investigated and a stability criterion for SFDSs is obtained, which is described by algebraic matrix equations. Finally, an example is given to illustrate the effectiveness of our method and results.     

Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations

Asymptotic stability and boundedness have been two of most popular topics in the study of stochastic functional differential equations (SFDEs) (see e.g. Appleby and Reynolds (2008), Appleby and Rodkina (2009), Basin and Rodkina (2008), Khasminskii (1980), Mao (1995), Mao (1997), Mao (2007), Rodkina and Basin (2007), Shu, Lam, and Xu (2009), Yang, Gao, Lam, and Shi (2009), Yuan and Lygeros (2005) and Yuan and Lygeros (2006)). In general, the existing results on asymptotic stability and boundedness of SFDEs require (i) the coefficients of the SFDEs obey the local Lipschitz condition and the linear growth condition; (ii) the diffusion operator of the SFDEs acting on a C^2,1-function be bounded by a polynomial with the same order as the C^2,1-function. However, there are many SFDEs which do not obey the linear growth condition. Moreover, for such highly nonlinear SFDEs, the diffusion operator acting on a C^2,1-function is generally bounded by a polynomial with a higher order than the C^2,1-function. Hence the existing criteria on stability and boundedness for SFDEs are not applicable and we see the necessity to develop new criteria. Our main aim in this paper is to establish new criteria where the linear growth condition is no longer needed while the up-bound for the diffusion operator may take a much more general form.     

Robust continuous-time smoothers without two-sided stochastic integrals

We consider the problem of fixed-interval smoothing of a continuous-time partially observed nonlinear stochastic dynamical system. Existing results for such smoothers require the use of two-sided stochastic calculus. The main contribution of the paper is to present a robust formulation of the smoothing equations. Under this robust formulation, the smoothing equations are nonstochastic parabolic partial differential equations (with random coefficients) and, hence, the technical machinery associated with two sided stochastic calculus is not required. Furthermore, the robust smoothed state estimates are locally Lipschitz in the observations, which is useful for numerical simulation. As examples, finite dimensional robust versions of the Benes and hidden Markov model smoothers and smoothers for piecewise linear dynamics are derived; these finite-dimensional smoothers do not involve stochastic integrals.     

On large deviations for Poisson stochastic integrals

We obtain asymptotically exact estimates for large deviations of Poisson stochastic integrals. We also find a region where such an integral can be approximated by the corresponding Gaussian random variable. In of all these results, we obtain nonasymptotic estimates for remainder terms.     

On the exponential stability in mean square of neutral stochastic functional differential equations

In [1 and 2], some efforts have been devoted to the investigation of exponential stability in mean square of neutral stochastic functional differential equations. However, the results derived there are either difficult to demonstrate in a straightforward way for practical situations or somewhat too restricted to be applied to general neutral stochastic functional differential equations, for instance, nonautonomous cases. In this paper, we shall establish some results which are more effective and relatively easy to verify to obtain the required stability.     

Application of the hybrid functions to solve neutral delay functional differential equations

This paper presents a computational technique for the solution of the neutral delay differential equations with state-dependent and time-dependant delays. The properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. The approach uses these properties together with the collocation points to reduce the main problems to systems of nonlinear algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.     

Attraction, stability and robustness for stochastic functional differential equations with infinite delay

This paper establishes the stochastic LaSalle theorem to locate limit sets for stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness, stability and robustness are obtained. To illustrate the applications of our results clearly, this paper considers a scalar stochastic integro-differential equation with infinite delay as an example.     

Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay as the result of the effects of stochastic perturbation to the deterministic functional differential equation , where is defined by x_t(θ)=x(t+θ),θ(−∞,0]. We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.     

Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay

This paper establishes the existence-and-uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate. This result may be used to examine almost sure robust stability. To illustrate our idea more carefully, we carefully discuss a scalar stochastic integro-differential equation with neutral type and its asymptotic stability, including the exponential stability and the polynomial stability.     

Markov-modulated stochastic recursive equations with applications to delay-tolerant networks

We investigate a class of Markov-modulated stochastic recursive equations. This class includes multi-type branching processes with immigration as well as linear stochastic equations. Conditions are established for the existence of a stationary solution and expressions for the first two moments of this solution are found. Furthermore, the transient characteristics of the stochastic recursion are investigated: we obtain the first two moments of the transient solution as well. Finally, to illustrate our approach, the results are applied to the performance evaluation of packet forwarding in delay-tolerant mobile ad-hoc networks.     

On stochastic Riccati equations for the stochastic LQR problem

This paper studies an approximation of stochastic Riccati equations for stochastic LQR problems some of which may be even with indefinite control weight costs.     

Representation of the bilinear system output by multiple stochastic integrals

Consideration was given to the scalar stochastic differential Ito equation whose drift and diffusion coefficients are affine functions of the phase coordinate. Its solution was represented in terms of the stochastic exponent which plays the part of the equation resolvent. Expansion of the stochastic exponent in series in the Hermit polynomials induces expansion of the solution of the bilinear equation in series in multiple stochastic integrals. Obtained were the moment characteristics of the stochastic integrals solving the problem of statistical analysis of the approximate solutions of the bilinear stochastic systems. A model of the financial (B, S)-market and its optimization in terms of a nonsquare criterion were considered by way of example of a bilinear system.