Singular perturbations in coupled stochastic differential equations.

摘要

In this thesis we study the coupled system of stochastic integral equations xet≜ x0+e0>t F&parl0;xes ,yes&parr0; ds+e1/20>t G&parl0;xes &parr0;dws, 0.0.1 yet≜ y0+0>tb&parl0; xes,y es&parr0;ds+ 0>ts&parl0;xe s,yes &parr0;bs, 0.0.2 in which &epsis; > 0 is a small parameter, {lcub}x&epsis;(t){rcub} is an $Rd$-valued slow process, and {lcub}y &epsis;(t){rcub} is an $RD$-valued fast process. Our general goal is to characterize asymptotic properties of the slow process {lcub}x&epsis; (t){rcub} over intervals of the form 0 ≤ t ≤ T/&epsis;, for a fixed constant T ∈ (0, ∞), as &epsis; → 0. The motivation for studying this question is a result of Khas'minskii (“On the Averaging Principle for Itô Stochastic Differential Equations”, Kybernetika , V. 4(3): 260–279, 1968 (Russian), also stated as Theorem 9.1 on page 264 of the book Random Perturbations of Dynamical Systems by Freidlin and Wentzel, Springer-Verlag, 1984), which basically goes as follows: suppose that the auxiliary stochastic differential equation dxt=b x,xt dt+sx,xt dbt 0.0.3 (which is really just (0.0.2), but with the slow variables x&epsis;(s) “frozen” at some fixed x ∈ $Rd$) is “stable”, in the sense that the Markov process arising from (0.0.3) has a unique invariant probability measure π _x, for each x ∈ $Rd$. Define the “averaged” drift Fx≜ RDF x,xdpx x, 0.0.4 and use this to write the “averaged” version of (0.0.1) name