Rafail Z. Khasminskii

On Averaging Principles: An Asymptotic Expansion Approach

This work is concerned with diffusion processes having fast and slow components. It was known that under suitable assumptions the slow component can be approximated by the Markov process with averaged characteristics. In this work, asymptotic expansions for the solutions of the Kolmogorov backward equations are constructed and justified. Certain probabilistic conclusions and examples are also provided.

Moment Lyapunov exponent and stability index for linear conservative system with small random perturbation

Asymptotic expansion series for the moment Lyapunov exponent and stability index are constructed and justified for the two-dimensional linear stochastic system close to a harmonic oscillator. As an example, a one-degree-of-freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained.

Stationary solutions of nonlinear stochastic evolution equations

General theorems concerning the existence and uniqueness of invariant measures are proved for a certain class of regular diffusion processes in Separable Banach spaces under some weak compactness and other conditions. Then, based on these theorems, some verifiable sufficient conditions are obtained to ensure the existence and uniqueness of an invariant distribution for the strong solution to some nonlinear evolution equations in a Hilbert space. The results are applied to certain monotone parabolic Ito equations as well as to the 2-D Navier-Stokes equations under random perturbations

Asymptotic series for singularly perturbed Kolmogorov-Fokker-Planck equations

We derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed Kolmogorov–Fokker–Planck equations. The model under consideration can be viewed as a Markov process having two time scales. One of them is a rapidly changing scale, and the other is a slowly varying one. The study is motivated by a wide range of applications involving singularly perturbed Markov processes in manufacturing systems, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. In this work, the asymptotic expansion is constructed explicitly. It is shown that the initial layer terms in the expansion decay at an exponential rate. Error bounds on the remainder terms also are obtained. The validity of the expansion is rigorously justified.

On transition densities of singularly perturbed diffusions with fast and slow components

We derive asymptotic properties of transition densities for singularly perturbed diffusion processes with fast and slow components. Our study focuses on the Kolmogorov–Fokker–Planck equations. The model can be viewed as a diffusion process having two time scales and is motivated by a wide variety of applications involving singularly perturbed Markov processes in manufacturing systems, homogenization, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. By virtue of the methods of matched singular perturbation, asymptotic expansion is constructed for the transition density. The expansion includes both regular part and boundary layer corrections. Detailed justification of the asymptotic expansion is given, and error bounds are also provided.

Asymptotic filtering for finite state Markov chains

Asymptotic formulae for the optimal filtering error for finite state space Markov chains observed in independent noise are presented. Asymptotically optimal simple filters, which do not depend on the transition rates of the chain, are also presented.

ASYMPTOTIC EXPANSION OF SHIP CAPSIZING IN RANDOM SEA WAVES-I. FIRST-ORDER APPROXIMATION

Abstract The first passage problem of ship non-linear roll oscillations in random sea waves is examined. The ship roll dynamics are described by a non-linear stochastic differential equation which includes non-linear wave drag force and non-linear restoring moment. The non-linear restoring moment is divided into a sine function plus a correction function. The unperturbed motion of the ship is studied as a classical pendulum problem in terms of elliptic functions. The mean exit time of the perturbed ship motion is described by Pontryagins partial differential equation. The method of asymptotic expansion is employed to solve this equation. Within the framework of first-order approximation, the analysis reduces the Pontryagin equation into a second-order linear differential equation with variable coefficients. These coefficients are functions of the energy level of the ship. The solution of this equation is obtained in a closed form and is found to be well behaved, with resolvable singularities. The dependence of the mean exit time on the initial energy level, non-linear drag coefficient, and excitation spectral density is graphically plotted. Second-order approximation is treated in Part II of this two-part paper.

On averaging principle for diffusion processes with null-recurrent fast component

An averaging principle is proved for diffusion processes of type (X[var epsilon](t),Y[var epsilon](t)) with null-recurrent fast component X[var epsilon](t). In contrast with positive recurrent setting, the slow component Y[var epsilon](t) alone cannot be approximated by diffusion processes. However, one can approximate the pair (X[var epsilon](t),Y[var epsilon](t)) by a Markov diffusion with coefficients averaged in some sense.

Estimation of parameters of linear homogeneous stochastic differential equations

In this paper we investigate the problem of parametric estimation for multidimensional linear autonomous homogeneous stochastic differential equations. We prove the Local Asymptotical Normality (LAN) property, find the Maximum Likelihood Estimator (MLE), and prove an asymptotical efficiency of MLE for bounded loss functions, when the observation time tends to infinity.

Stability of Stochastic Differential Equations

In Chap. 1 we studied problems of stability under random perturbations of the parameters. We noted there that no significant results can be expected unless the random perturbations possess sufficiently favorable mixing properties. Fortunately, in practical applications one may often assume that the “noise” has a “short memory interval.” The natural limiting case of such noise is of course white noise. Thus it is very important to study the stability of solutions of Ito equations since this is equivalent to the study of stability of systems perturbed by white noise. Generalization of well known results on stability and instability for the deterministic ODE in terms of the Lyapunov functions are proven for SDE. Conditions for stability and instability of moments are also proven.