Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author's earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs

Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations

In this paper, the Euler–Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations. Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method

Classication of the asymptotic behaviour of solutions of stochastic differential equations with state independent noise

We investigate the asymptotic behaviour of solution of dierential equation with state-independent perturbation. The dierential equation studied is a perturbed version of a globally stable autonomous equation with unique equilibrium where the diffusion coefficient is independent of the state. Perturbed differential equation is widely applied to model natural phenomena, in Finance, Engineering, Physics and other disciplines. Real-world processes are often subjected to interference in the form of random external perturbations. This could lead to a dramatic effect on the behaviour of these processes. Therefore it is important to analyse these equations. We start by considering an additive deterministic perturbation in Chapter 1. It is assumed that the restoring force is asymptotically negligible as the solution becomes large, and that the perturbation tends to zero as time becomes indefinitely large. It is shown that solutions are always locally stable, and that solutions either tend to zero or to infinity as time tends to infinity. In Chapter 2 and 4, we each explore a linear and nonlinear equation with stochastic perturbation in finite dimensions. We find necessary and sufficient conditions on the rate of decay of the noise intensity for the solution of the equations to be globally asymptotically stable, bounded, or unstable. In Chapter 3 we concentrate on a scalar nonlinear stochastic differential equation. As well as the necessary and sufficient condition, we also explore the simple sufficient conditions and the connections between the conditions which characterise the various classes of long-run behaviour. To facilitate the analysis, we investigate using Split-Step method the difference equations both in the scalar case and the finite dimensional case in Chapter 5 and 6. We can mimic the exact asymptotic behaviour of the solution of the stochastic differential equation under the same conditions in discrete time

Highly nonlinear stochastic and deterministic differential equations with time-varying shocks: asymptotic behaviour and numerical analysis

This thesis concerns the asymptotic behaviour for nonlinear differential equations, and also considers how this behaviour can be recovered by appropriate numerical schemes. In particular, perturbed equations are studied, where the equation without perturbations has known asymptotic behaviour. The restoring force is generally not of linear order close to the equilibrium, and the perturbation, which is time-varying forcing function, may be very irregular. The thesis addresses three questions: first, what conditions on the forcing function characterize the case when the rate of decay of the solution of the unperturbed equation is preserved, and what is the decay rate for more slowly decaying forcing functions? Equations for which there is faster than power decay in the solution of the unperturbed equation are considered. This analysis involves generalising the class of regularly varying functions, as well generalising the notion of the Liapunov exponent to equations without leading order linear terms. Perturbation theorems, for which the decay rates of the unperturbed solutions are directly recovered, are also given. Second, we prove that continuous time behaviour can be reproduced numerically. This is done when faster-than-power law, but slower than exponential, decay occurs. A semi-implicit method is used to cope with strong global nonlinearities. If the nonlinearity is smaller than linear order close to equilibrium, a fixed step-size scheme recovers the asymptotic behaviour. Thirdly, it can be shown that the results can be applied to stochastically forced equations if the shocks have state-independent intensity. Numerical results are also presented, and the method in the deterministic case can be adapted to deal with the asymptotic behaviour of the perturbation, as well as the nonlinearity