Averaging of Singularly Perturbed Controlled Stochastic Differential Equations

Appl Math Optim 56:169209 (2007)

DOI: 10.1007/s00245-007-0893-6

Averaging of Singularly Perturbed Controlled Stochastic Differential Equations

Vivek Borkar1 and Vladimir Gaitsgory2

1School of Technology and Computer Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, [email protected]

2School of Mathematics, University of South Australia,

Mawson Lakes Campus, Mawson Lakes, SA 5095, [email protected]

Abstract. An averaged system to approximate the slow dynamics of a two time-scale nonlinear stochastic control system is introduced. Validity of the approximation is established. Special cases are considered to illustrate the general theory.

Key Words. Singularly perturbed controlled stochastic differential equations, Occupational measures, Averaging method, Limit occupational measures sets, Approximation of slow motions.

AMS Classication. 34E15, 34C29, 34A60, 93C70.

1. Introduction

In this paper we consider a system of nonlinear singularly perturbed (SP) controlled stochastic differential equations (CSDE). A small parameter > 0 is introduced in the system in such a way that the state variables are decomposed into a group of slow variables, which change their values with rates of the order O(1), and a group of fast ones, which change their values with rates of the order O(1/).

Singularly perturbed problems of control and optimization have been studied intensively in both deterministic and stochastic settings (see [1][12], [14], [20][25], [30],[31], [33][36], [38][46], [48], [49], [52][58], [61][63], [65], [67] and references

Vivek Borkar was partly supported by Grant III.5(157)/99-ET from the Department of Science and Technology, Govenment of India. The research undertaken by Vladimir Gaitsgory was supported by Australian Research Council Discovery Grants DP0346099 and DP0664330.

2007 Springer Science+Business Media, Inc.

170 V. Borkar and V. Gaitsgory

therein for a sample of the literature), with SP CSDE being specically addressed in [2],[3], [12], [14], [20], [40], [41] and [48].

Originally, the most common approaches to SP control systems, especially in the

deterministic case, were related to a so-called reduction technique based on equating of the small parameter to zero (that is, establishing results in the spirit of Tichonovs theorem; see [14], [22], [40], [41], [45], [46], [48], [54], [58], [62] and [65]). Later, it was realized that, while being very efcient in many important special classes of problems, the reduction technique approaches may not lead to a correct approximation in dealing with general nonlinear SP control systems (see, e.g., [5],...