Appl Math Optim 56:169–209 (2007)
2007 Springer Science+Business Media, Inc.
Averaging of Singularly Perturbed Controlled Stochastic
and Vladimir Gaitsgory
School of Technology and Computer Science, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India
School of Mathematics, University of South Australia,
Mawson Lakes Campus, Mawson Lakes, SA 5095, Australia
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 Classiﬁcation. 34E15, 34C29, 34A60, 93C70.
In this paper we consider a system of nonlinear singularly perturbed (SP) controlled
stochastic differential equations (CSDE). A small parameter ε>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/ε).
Singularly perturbed problems of control and optimization have been studied inten-
sively in both deterministic and stochastic settings (see –, , –, ,
, –, –, , , –, –, ,  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.