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Further Calculations for the McKean Stochastic Game for a Spectrally Negative Lévy Process: From a Point to an Interval

Part of: Game theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

E. J. Baurdoux  and
K. Van Schaik
E. J. Baurdoux*
Affiliation:
London School of Economics
K. Van Schaik*
Affiliation:
University of Bath
*
∗Postal address: Department of Statistics, London School of Economics, Houghton Street, London, WC2A 2AE, UK. Email address: e.j.baurdoux@lse.ac.uk
∗∗Current address: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK. Email address: kees.vanschaik@manchester.ac.uk
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Abstract

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Following Baurdoux and Kyprianou (2008) we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Lévy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular, we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and ‘thickens’ to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functions for the general case and in terms of polynomials for a specific jump diffusion case are provided.

Keywords

Stochastic gameoptimal stoppingLévy processfluctuation theory

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A15: Stochastic games
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 200 - 216
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

This author gratefully acknowledges being supported by a postdoctoral grant from the AXA Research Fund.

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