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Optimal stopping with random intervention times

Part of: Markov processes Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  19 February 2016

Paul Dupuis  and
Hui Wang
Paul Dupuis*
Affiliation:
Brown University
Hui Wang*
Affiliation:
Brown University
*
∗ Postal address: Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA.
∗ Postal address: Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA.
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Abstract

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞2 otherwise, both contradicting the usual 𝒞1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

Keywords

Optimal stoppingvariational inequalitystochastic controlPoisson process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 93E20: Optimal stochastic control
Secondary: 60J65: Brownian motion 60J75: Jump processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 141 - 157
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Research supported in part by the National Science Foundation (NSF-DMS-0072004) and the Army Research Office (ARO-DAAD19-99-1-0223).

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