Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation

We consider a non-homogeneous nonlinear stochastic difference equation X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important special case X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables \xi_n. We establish results on \as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise \xi_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning the rate of decay of X_n to zero.Comment: 22 pages; corrected more typos, fixed LaTeX macro

Stabilisation and destabilisation of nonlinear differential equations by noise

This paper considers the stabilisation and destabilisa- tion by a Brownian noise perturbation which preserves the equilibrium of the ordinary dierential equation x0(t) = f(x(t)). In an extension of earlier work, we lift the restriction that f obeys a global linear bound, and show that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g(X(t)) dB(t) either stabilises an unstable equilibrium, or destabilises a stable equilibrium. When the equilibrium of the deterministic equation is non{hyperbolic, we show that a non{hyperbolic perturbation suffices to change the stability properties of the solution.

On the local dynamics of polynomial difference equations with fading stochastic perturbations

We examine the stability-instability behaviour of a polynomial difference equa- tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis- cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region

On the oscillation of solutions of stochastic difference equations.

This paper considers the pathwise oscillatory behaviour of the scalar nonlinear stochastic dif- ference equation X(n + 1) = X(n) − F (X(n)) + G(n, X(n))ξ(n + 1), n = 0, 1, . . . , with non-random initial value X0 . Here (ξ(n))n≥0 is a sequence of independent random variables with zero mean and unit variance. The functions f : R → R and g : R → R are presumed to be continuous

Stabilization of cycles with stochastic prediction-based and target-oriented control

We stabilize a prescribed cycle or an equilibrium of a difference equation using pulsed stochastic control. Our technique, inspired by Kolmogorov's law of large numbers, activates a stabilizing effect of stochastic perturbation and allows for stabilization using a much wider range for the control parameter than would be possible in the absence of noise. Our main general result applies to both prediction-based and target-oriented controls. This analysis is the first to make use of the stabilizing effects of noise for prediction-based control; the stochastic version has previously been examined in the literature, but only the destabilizing effect of noise was demonstrated. A stochastic variant of target-oriented control has never been considered, to the best of our knowledge, and we propose a specific form that uses a point equilibrium or one point on a cycle as a target. We illustrate our results numerically on the logistic, Ricker, and Maynard Smith models from population biology