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Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes

Part of: Functional-differential and differential-difference equations Stochastic processes Mathematical economics Markov processes

Published online by Cambridge University Press:  14 July 2016

Pavel V. Gapeev
Pavel V. Gapeev*
Affiliation:
WIAS and Russian Academy of Sciences
*
∗Postal address: Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, D-10117 Berlin, Germany. Email address: gapeev@wias-berlin.de and p.gapeev@lse.ac.uk
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Abstract

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In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

Keywords

Discounted optimal stopping problemBrownian motioncompound Poisson processmaximum processintegro-differential free-boundary problemcontinuous and smooth fitnormal reflectionchange-of-variable formula with local time on surfacesperpetual American lookback option

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 34K10: Boundary value problems 91B70: Stochastic models 60J60: Diffusion processes 60J75: Jump processes
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 713 - 731
Copyright
Copyright © Applied Probability Trust 2007 

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