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A class of singular stochastic control problems

Published online by Cambridge University Press:  01 July 2016

Ioannis Karatzas
Ioannis Karatzas*
Affiliation:
Columbia University
*
∗Postal address: Department of Mathematical Statistics, Columbia University, New York, NY 10027, U.S.A.
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Abstract

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

Keywords

STOCHASTIC CONTROLOPTIMAL STOPPINGREFLECTED BROWNIAN MOTIONFREE-BOUNDARY PROBLEMS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 2 , June 1983 , pp. 225 - 254
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

This paper was presented at the second Bad Honnef Workshop on Stochastic Differential Systems, University of Bonn, West Germany, 28 June–2 July 1982.

Research partly supported by NSF Grant MCS 81–03435.

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