On input-to-state stability of stochastic retarded systems with Markovian switching

This note develops a Razumikhin-type theorem on pth moment input-to-state stability of hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), which is an improvement of an existing result. An application to hybrid stochastic delay systems verifies the effectiveness of the improved result

Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity

Retarded stochastic differential equations (SDEs) constitute a large collection of systems arising in various real-life applications. Most of the existing results make crucial use of dissipative conditions. Dealing with "pure delay" systems in which both the drift and the diffusion coefficients depend only on the arguments with delays, the existing results become not applicable. This work uses a variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. This paper establishes existence and uniqueness of stationary distributions for retarded SDEs that need not satisfy dissipative conditions. The retarded SDEs considered in this paper also cover SDEs of neutral type and SDEs driven by L\'{e}vy processes that might not admit finite second moments.Comment: page 2

Exponential Mixing for Retarded Stochastic Differential Equations

In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded SDEs by the Arzel\`{a}--Ascoli tightness characterization of the space \C equipped with the uniform topology (b) neutral SDEs with continuous sample paths by a generalized Razumikhin-type argument and a stability-in-distribution approach and (c) jump-diffusion retarded SDEs by the Kurtz criterion of tightness for the space \D endowed with the Skorohod topology.Comment: 20 page

Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching

The main aim of this paper is to discuss the almost surely asymptotic stability of the neutral stochastic differential delay equations (NSDDEs) with Markovian switching. Linear NSDDEs with Markovian switching and nonlinear examples will be discussed to illustrate the theory

A new criterion on stability in distribution for a hybrid stochastic delay differential equation

A new sufficient condition for stability in distribution of a hybrid stochastic delay differential equation (SDDE) has been proposed. In the new criterion leading to stability for an SDDE, its main component only depends on the coefficients of a corresponding SDE without delay. The Lyapunov method is applied to find an upper bound, so that the SDDE is stable in distribution if the delay is less than the upper bound. Also, the criterion shows that delay terms can be impetuses toward the stability in distribution

Invariant probability measures for path-dependent random diffusions

In this work, we are concerned with path-dependent random diffusions. Under certain ergodic condition, we show that the path-dependent random diffusion under consideration has a unique invariant probability measure and converges exponentially to its equilibrium under the Wasserstein distance. Also, we demonstrate that the time discretization of the path-dependent random diffusion involved admits a unique (numerical) invariant probability measure and preserves the corresponding ergodic property when the step size is sufficiently small. Moreover, we provide an estimate on the exponential functional of the discrete observation for a Markov chain, which may be interesting by itself