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Discretionary stopping of one-dimensional Itô diffusions with a staircase reward function

Part of: Applications Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Anne Laure Bronstein ,
Lane P. Hughston ,
Martijn R. Pistorius  and
Mihail Zervos
Anne Laure Bronstein
Affiliation:
King's College London
Lane P. Hughston*
Affiliation:
King's College London
Martijn R. Pistorius*
Affiliation:
King's College London
Mihail Zervos
Affiliation:
King's College London
*
∗∗Postal address: Department of Mathematics, King's College London, The Strand, London WC2R 2LS, UK.
∗∗Postal address: Department of Mathematics, King's College London, The Strand, London WC2R 2LS, UK.
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Abstract

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We consider the problem of optimally stopping a general one-dimensional Itô diffusion X. In particular, we solve the problem that aims at maximising the performance criterion Ex[exp(-∫0τr(Xs)ds)f(Xτ)] over all stopping times τ, where the reward function f can take only a finite number of values and has a ‘staircase’ form. This problem is partly motivated by applications to financial asset pricing. Our results are of an explicit analytic nature and completely characterise the optimal stopping time. Also, it turns out that the problem's value function is not C1, which is due to the fact that the reward function f is not continuous.

Keywords

Optimal stoppingAmerican optionprinciple of smooth fitlocal time

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 93E03: Stochastic systems, general 93E20: Optimal stochastic control
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 984 - 996
Copyright
© Applied Probability Trust 2006 

Footnotes

∗

Current address: Laboratoire de Probabilités et Modèles Aléotoires, Université Paris 6, 175 rue du Chevaleret, Paris, 75013, France. Email address: albronstein@hotmail.com

∗∗∗∗∗

Current address: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: m.zervos@lse.ac.uk

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