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Stationary stochastic control for Itô processes

Part of: Stochastic systems and control Stochastic analysis Stochastic processes

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: weera@math.iastate.edu
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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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References

Bather, J. A. and Chernoff, H. (1967). Sequential decisions in the control of a spaceship (finite fuel). J. Appl. Prob. 4, 584–604.Google Scholar
Beneš, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some solvable stochastic control problems. Stochastics 4, 39–83.Google Scholar
Borkar, V. S. and Ghosh, M. K. (1988). Ergodic control of multidimensional diffusions I: The existence results. SIAM J. Control Optimization 26, 112–126.Google Scholar
Chiarolla, M. B. and Haussmann, U. G. (1998). Controlling inflation: the infinite horizon case. SIAM J. Control Optimization 36, 1099–1132.Google Scholar
Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, New York.Google Scholar
Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.Google Scholar
Haussmann, U. G. (1986). A Stochastic Maximum Principle for Optimal Control of Diffusions (Pitman Res. Notes Math. 151). Longman, Harlow.Google Scholar
Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225–254.Google Scholar
Kurtz, T. G. and Stockbridge, R. H. (1998). Existence of Markov controls and characterization of optimal Markov controls. SIAM J. Control Optimization 36, 609–653.Google Scholar
Kushner, H. J. (1978). Optimality conditions for the average cost per unit time problem with a diffusion model. SIAM J. Control Optimization 16, 330–346.Google Scholar
Meyer, P. A. (1974). Un cours sur les intégrales stochastiques. In Seminaire de Probabilités X (Lecture Notes Math. 511). Springer, New York.Google Scholar
Musiela, M. and Rutkowski, M. (1997). Mathematical Methods in Financial Modelling. Springer, New York.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations: A Unified Approach. Springer, New York.Google Scholar
Protter, M. H. and Weinberger, H. F. (1984). Maximum Principles in Differential Equations, 2nd edn. Springer, New York.Google Scholar
Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transaction costs. Ann. Appl. Prob. 4, 609–692.Google Scholar
Taksar, M., Klass, M. J. and Assaf, D. (1988). A diffusion model for optimal portfolio selection in the presence of brokerage fees. Math. Operat. Res. 13, 277–294.Google Scholar