Singularly Perturbed Linear Stochastic Ordinary Differential Equations

Abstract

Singularly perturbed linear differential equations with random forcing functions have recently been studied as models of control and filtering systems. The analysis in these studies has been somewhat formal, and important properties of the boundary layer behavior have been neglected as a consequence. In the present paper we examine the asymptotic analysis of systems of this type using some limit theorems from the theory of stochastic processes. We show that a natural separation of time scales occurs between the “outer” and “boundary layer” solutions and their respective stochastic fluctuations. A total of four basic time scales is necessary for a complete description of the solution. The separation of scales is characterized by a parameter $\varepsilon $ related to the time constant of the parasitics (fast subsystem) and the correlation time (inverse of bandwidth) of the stochastic fluctuations. We show that certain diffusion processes may be identified as the natural limits as $\varepsilon \to 0$ of the “outer solution” and the “boundary layer correction” of the original system.