Open Access
February 1999 Control and stopping of a diffusion process on an interval
Ioannis Karatzas, William D. Sudderth
Ann. Appl. Probab. 9(1): 188-196 (February 1999). DOI: 10.1214/aoap/1029962601

Abstract

Consider a process X()=X(t),0t< which takes values in the interval I=(0,1), satisfies a stochastic differential equation dX(t)=β(t)dt+σ(t)dW(t),X(0)=xϵI and, when it reaches an endpoint of the interval I, it is absorbed there. Suppose that the parameters β and σ are selected by a controller at each instant tϵ[0,) from a set depending on the current position. Assume also that the controller selects a stopping time τ for the process and seeks to maximize Eu(X(τ)), where u:[0,1] is a continuous "reward" function. If λ:=infxϵI:u(x)=maxu and ρ:=supxϵI:u(x)=maxu, then, to the left of λ, it is best to maximize the mean-variance ratio (β/σ2) or to stop, and to the right of ρ, it is best to minimize the ratio (β/σ2) or to stop. Between λ and ρ, it is optimal to follow any policy that will bring the process X() to a point of maximum for the function u() with probability 1, and then stop.

Citation

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Ioannis Karatzas. William D. Sudderth. "Control and stopping of a diffusion process on an interval." Ann. Appl. Probab. 9 (1) 188 - 196, February 1999. https://doi.org/10.1214/aoap/1029962601

Information

Published: February 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0938.93067
MathSciNet: MR1682584
Digital Object Identifier: 10.1214/aoap/1029962601

Subjects:
Primary: 60G40 , 93E20
Secondary: 60D60 , 62L15

Keywords: one-dimensional diffusions , Optimal stopping , Stochastic control

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.9 • No. 1 • February 1999
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