Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T01:32:04.011Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  14 July 2016

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Papers
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: [email protected]

∗∗∗∗∗

Current address: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: [email protected]

References

Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173212.Google Scholar
Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468519.Google Scholar
Itô, K. and McKean, H. P. Jr. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Karatzas, I. and Sudderth, W. D. (1999). Control and stopping of a diffusion process on an interval. Ann. Appl. Prob. 9, 188196.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion, 2nd edn. Springer, Berlin.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus. Cambridge University Press.Google Scholar
Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85101.Google Scholar
Zervos, M. (2003). A problem of sequential entry and exit decisions combined with discretionary stopping. SIAM J. Control Optimization 42, 397421.Google Scholar