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Finite-time optimal control of a process leaving an interval

Part of: Stochastic processes Markov processes Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Douglas W. Mcbeth  and
Ananda P. N. Weerasinghe
Douglas W. Mcbeth*
Affiliation:
Iowa State University
Ananda P. N. Weerasinghe*
Affiliation:
Iowa State University
*
∗Postal address: Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, IA 50011, USA.
∗∗Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
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Abstract

Consider the optimal control problem of leaving an interval (– a, a) in a limited playing time. In the discrete-time problem, a is a positive integer and the player's position is given by a simple random walk on the integers with initial position x. At each time instant, the player chooses a coin from a control set where the probability of returning heads depends on the current position and the remaining amount of playing time, and the player is betting a unit value on the toss of the coin: heads returning +1 and tails − 1. We discuss the optimal strategy for this discrete-time game. In the continuous-time problem the player chooses infinitesimal mean and infinitesimal variance parameters from a control set which may depend upon the player's position. The problem is to find optimal mean and variance parameters that maximize the probability of leaving the interval [— a, a] within a finite time T > 0.

Keywords

STOCHASTIC OPTIMAL CONTROLGAMBLING PROBLEMSBOLD PLAYPARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 714 - 728
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Fleming, W. H. and Soner, H. M. (1993) Controlled Markov Processes and Viscosity Solutions. Springer, Berlin.Google Scholar
[2] Friedman, A. (1983) Partial Differential Equations of Parabolic Type. Krieger, Malabar, FL.Google Scholar
[3] Heath, D. and Kertz, R. P. (1988) Leaving an interval in limited playing time. Adv. Appl. Prob. 20, 635–645.CrossRefGoogle Scholar
[4] Heath, D., Orey, S., Pestien, V. and Sudderth, W. (1987) Minimizing or maximizing the expected time to reach zero. SIAM J Control Opt. 25, 195–205.CrossRefGoogle Scholar
[5] Heath, D. and Sudderth, W. (1974) Continuous-time gambling problems. Adv. Appl. Prob. 6, 651–665.Google Scholar
[6] Ikeda, N. and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam.Google Scholar
[7] Karatzas, I. and Shreve, S. E. (1988) Brownian Motion and Stochastic Calculus. Springer, Berlin.CrossRefGoogle Scholar
[8] Ladyzenskaja, O. A., Solonnikov, V. A. and Ural'Ceva, N. N. (1968) Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, RI.Google Scholar
[9] Lieberman, G. (1986) Mixed boundary value problems for elliptic and parabolic differential equations of second-order. J. Math. Anal. Appl. 113, 422–440.CrossRefGoogle Scholar
[10] Pestien, V. and Sudderth, W. (1983) Continuous-time red and black: How to control a diffusion to a goal. Math. Operat. Res. 10, 599–611.CrossRefGoogle Scholar
[11] Protter, M. H. and Weinberger, H. F. (1984) Maximum Principles in Differential Equations. Springer, Berlin.CrossRefGoogle Scholar
[12] Spencer, J. (1986) Balancing vectors in the max norm. Combinatorica 6, 55–65.Google Scholar
[13] Weerasinghe, A. P. N. (1992) A finite-fuel stochastic control problem on a finite time horizon. SIAM J. Control Opt. 30, 1395–1408.Google Scholar