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Stationary stochastic control for Itô processes

Part of: Stochastic analysis Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  19 February 2016

Ananda P. N. Weerasinghe
Ananda P. N. Weerasinghe*
Affiliation:
Iowa State University
*
∗ Postal address: Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA. Email address: [email protected]
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Abstract

Consider a real-valued Itô process X(t) = x + ∫0tμ(s)ds + ∫0tσ(s)dW(s) + A(t) driven by a Brownian motion {W(t) : t > 0}. The controller chooses the real-valued progressively measurable processes μ, σ and A subject to constraints |μ(t)| ≤ μ0(X(t-)) and |σ(t)| ≥ σ0(X(t-)), where the functions μ0 and σ0 are given. The process A is a bounded variation process and |A|(t) represents its total variation on [0,t]. The objective is to minimize the long-term average cost lim supT→∞(1/T)E[|A|(T) + ∫0Th(X(s))ds], where h is a given nonnegative continuous function. An optimal process X* is determined. It turned out that X* is a reflecting diffusion process whose state space is a finite interval [a*, b*]. The optimal drift and diffusion controls are explicitly derived and the optimal bounded variation process A* is determined in terms of local-time processes of X* at the points a* and b*.

Keywords

Stochastic differential equationsreflecting diffusion processeslocal time processesstationary control

MSC classification

Primary: 93E20: Optimal stochastic control 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60H10: Stochastic ordinary differential equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 128 - 140
Copyright
Copyright © Applied Probability Trust 2002 

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