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OPTIMAL STOPPING FOR THE LAST EXIT TIME

Part of: Stochastic processes Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  04 October 2018

DAN REN Open the ORCID record for DAN REN [Opens in a new window]
DAN REN*
Affiliation:
Department of Mathematics, University of Dayton, Dayton, OH 45469, USA email [email protected]
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Abstract

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Given a one-dimensional downwards transient diffusion process $X$, we consider a random time $\unicode[STIX]{x1D70C}$, the last exit time when $X$ exits a certain level $\ell$, and detect the optimal stopping time for it. In particular, for this random time $\unicode[STIX]{x1D70C}$, we solve the optimisation problem $\inf _{\unicode[STIX]{x1D70F}}\mathbb{E}[\unicode[STIX]{x1D706}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70C})_{+}+(1-\unicode[STIX]{x1D706})(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70F})_{+}]$ over all stopping times $\unicode[STIX]{x1D70F}$. We show that the process should stop optimally when it runs below some fixed level $\unicode[STIX]{x1D705}_{\ell }$ for the first time, where $\unicode[STIX]{x1D705}_{\ell }$ is the unique solution in the interval $(0,\unicode[STIX]{x1D706}\ell )$ of an explicitly defined equation.

Keywords

optimal stoppingrandom timedownwards transient diffusionlast exit time

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62M20: Prediction 60J65: Brownian motion
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 148 - 160
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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