arXiv subjects
Publications and source records attributed to Alexandra Rodkina.
In this paper we determine the exact rate of growth of the solution of a deterministic delay differential equation in which the delayed term is regularly varying at infinity and dominates, and determine criteria to characterise this dominance. The preservation of growth rates using a uniform step size Euler scheme is also discussed.
In this paper we classify the pathwise asymptotic behaviour of the discretisation of a general autonomous scalar differential equation which has a unique and globally stable equilibrium. The underlying continuous equation is subjected to a stochastic perturbation whose intensity is state--independent. In the main result, it is shown that when the split--step--method is applied to the resulting stochastic differential equation, and the stochastic intensity is decreasing, the solutions of the discretised equation inherit the asymptotic behaviour of the continuous equation, regardless of whether the continuous equation has stable, bounded but unstable, or unbounded solutions, provided the step size is chosen sufficiently small.
We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity for the solution. In particular, we found sufficient conditions for existence of a periodic process such that the solution converges to it, including almost surely convergence.
We consider a non-homogeneous nonlinear stochastic difference equation X_{n+1} = X_n (1 + f(X_n)ξ_{n+1}) + S_n, and its important special case X_{n+1} = X_n (1 + ξ_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables ξ_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 ξ_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.