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 C2,1-function be bounded by a polynomial with the same order as the C2,1-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 C2,1-function is generally bounded by a polynomial with a higher order than the C2,1-function. Hence the existing criteria on stability and boundedness for SFDEs are not applicable andwesee 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

Delay Feedback Control for Switching Diffusion Systems Based on Discrete-Time Observations

Since response lags are required by most of physical systems and play a key role in the feedback control, the aim of this paper is to design delay feedback control functions based on the discrete-time observations of the system states and the Markovian states in order for the controlled switching diffusion system (SDS) to be exponentially stable in $p$th moment and probability one as well as stable in H_\8. The main methods are the strong ergodicity theory of Markov chains and asymptotic analysis of stochastic functional differential equations (SFDEs). For the sake of saving time and costs the feedback control based on discrete-time observations is used to stabilize the switching diffusion systems.The designed control principles are implementable to stablize quasi-linear and highly nonlinear SDSs. For quasi-linear SDSs the criteria are sharp that under the control with high strength the controlled SDSs will be stable (bounded) while under the weaker control they will be unstable (unbounded) in mean square. The sample and moment Lyapunov exponents are estimated which have close relationship with the time delays

2013 (Spring)

Abstracts of the talks given at the 2013 Spring Colloquium

Competitive Lotka-Volterra Population Dynamics with Jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) We discuss the uniform boundedness of $p$th moment with $p>0$ and reveal the sample Lyapunov exponents; (c) Using a variation-of-constants formula for a class of SDEs with jumps, we provide explicit solution for 1-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $n$-dimensional model.Comment: 25 page

Asymptotic stability and boundedness of stochastic functional differential equations with Markovian switching

This paper is concerned with the boundedness, exponential stability and almost sure asymptotic stability of stochastic functional differential equations (SFDEs) with Markovian switching. The key technique used is the method of multiple Lyapunov functions. We use two auxiliary functions to dominate the corresponding different Lyapunov function in different mode while the diffusion operator in different model is controlled by other multiple auxiliary functions. Our conditions on the diffusion operator are weaker than those in the related existing works

Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models

Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients

This paper discusses exponential stability of solutions for highly nonlinear hybrid pantograph stochastic differential equations (PSDEs). Two criteria are proposed to guarantee exponential stability of the solution. The first criterion is a Khasminskii-type condition involving general Lyapunov functions. The second is developed on coefficients of the equation in virtue of M-matrix techniques. Based on the second criterion, robust stability of a perturbed hybrid PSDE is also investigated. The theory shows how much an exponentially stable hybrid PSDE can tolerate to remain stable