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Some control problems with random intervention times

Part of: Stochastic systems and control Markov processes

Published online by Cambridge University Press:  01 July 2016

Hui Wang
Hui Wang*
Affiliation:
Brown University
*
∗ Postal address: Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA. Email address: huiwang@cfm.brown.edu
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Abstract

We consider the problem of optimally tracking a Brownian motion by a sequence of impulse controls, in such a way as to minimize the total expected cost that consists of a quadratic deviation cost and a proportional control cost. The main feature of our model is that the control can only be exerted at the arrival times of an exogenous uncontrolled Poisson process (signal). In other words, the set of possible intervention times are discrete, random and determined by the signal process (not by the decision maker). We discuss both the discounted problem and the ergodic problem, where explicit solutions can be found. We also derive the asymptotic behavior of the optimal control policies and the value functions as the intensity of the Poisson process goes to infinity, or roughly speaking, as the set of admissible controls goes from the discrete-time impulse control to the continuous-time bounded variation control.

Keywords

impulse controlvariational inequalitygeneralized Ito formula

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60J65: Brownian motion 60J75: Jump processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 404 - 422
Copyright
Copyright © Applied Probability Trust 2001 

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