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Optimal stopping for the exponential of a Brownian bridge

Part of: Stochastic processes Markov processes Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  04 May 2020

Tiziano de Angelis Open the ORCID record for Tiziano de Angelis [Opens in a new window]  and
Alessandro Milazzo Open the ORCID record for Alessandro Milazzo [Opens in a new window]
Tiziano de Angelis*
Affiliation:
University of Leeds
Alessandro Milazzo*
Affiliation:
Imperial College London
*
*Postal address: School of Mathematics, University of Leeds, Woodhouse Lane, LS2 9JTLeeds, UK. Email address: t.deangelis@leeds.ac.uk
**Postal address: Department of Mathematics, Imperial College London, 16-18 Princess Gardens, SW7 1NELondon, UK. Email address: a.milazzo16@imperial.ac.uk
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Abstract

We study the problem of stopping a Brownian bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.

Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

Keywords

optimal stoppingBrownian bridgefree boundary problemsregularity of value functioncontinuous boundarybond/stock selling

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J65: Brownian motion 35R35: Free boundary problems
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 361 - 384
Copyright
© Applied Probability Trust 2020

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