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Excursions height- and length-related stopping times, and application to finance

Part of: Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  01 July 2016

Laurent Gauthier
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Abstract

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesney et al. (1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

Keywords

Parisian optionsBrownian excursionsreal options

MSC classification

Primary: 60J65: Brownian motion 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 91B06: Decision theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 846 - 868
Copyright
Copyright © Applied Probability Trust 2002 

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References

Abel, A. B. and Eberly, J. C. (1996). Optimal investment with costly reversibility. Rev. Econom. Stud. 63, 581–593.Google Scholar
Biane, P. and Yor, M. (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. 111, 23–101.Google Scholar
Borodin, A. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.Google Scholar
Chesney, M., Jeanblanc, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165–184.Google Scholar
Dixit, A. K. and Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton University Press.Google Scholar
Gauthier, L. (2002). Hedging entry and exit decisions: activating and deactivating barrier options. J. Appl. Math. Decision Sci. 6, 51–70.Google Scholar
Gauthier, L. and Morellec, E. (2001). Investment under uncertainty with implementation delay. In Real Options and Investment under Uncertainty, eds Schwartz, E. S. and Trigeorgis, L., MIT Press, Cambridge, MA.Google Scholar
He, H. and Pindyck, R. S. (1992). Investment in flexible production capacity. J. Econom. Dynamics Control 16, 575–599.Google Scholar
Imhof, J.-P. (1984). Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Prob. 21, 500–510.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin.Google Scholar
McDonald, R. and Siegel, D. (1986). The value of waiting to invest. Quart. J. Econom. 101, 707–728.Google Scholar
Majd, S. and Myers, S. C. (1990). Abandonment value and project life. Adv. Futures Options Res. 4, 1–21.Google Scholar
Mauer, D. C. and Ott, S. H. (1995). Investment under uncertainty: the case of replacement investment decisions. J. Financial Quantitative Anal. 30, 581–605.Google Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.Google Scholar
Trigeorgis, L. (1996). Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press, Cambridge, MA.Google Scholar
Yor, M. (1993). Some Aspects of Brownian Motion. Part I: Some Special Functionals. Birkhäuser, Basel.Google Scholar
Yor, M. (1997). Local Times and Excursion for Brownian Motion: a Concise Introduction. Lecciones in Matematicas, Universidad Central de Venezuela.Google Scholar
Yor, M. (1997b). Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems. Birkhäuser, Basel.Google Scholar