Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-27T20:54:44.548Z Has data issue: false hasContentIssue false
  • English
  • Français

Integral functionals under the excursion measure

Part of: Markov processes

Published online by Cambridge University Press:  04 May 2020

Maciej Wiśniewolski
Maciej Wiśniewolski*
Affiliation:
University of Warsaw
*
*Postal address: Institute of Mathematics, University of Warsaw Banacha 2, 02-097 Warszawa, Poland. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

A new approach to the problem of finding the distribution of integral functionals under the excursion measure is presented. It is based on the technique of excursion straddling a time, stochastic analysis, and calculus on local time, and it is done for Brownian motion with drift reflecting at 0, and under some additional assumptions for some class of Itó diffusions. The new method is an alternative to the classical potential-theoretic approach and gives new specific formulas for distributions under the excursion measure.

Keywords

Brownian motion with driftexcursions of Markov processexcursion measurefirst hitting time of 0Itó diffusion

MSC classification

Primary: 60J45: Probabilistic potential theory
Secondary: 60J55: Local time and additive functionals 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 137 - 155
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bertoin, J. (1999). Subordinators: examples and applications. In Lectures on Probability Theory and Statistics, ed. P. Bernard, Lecture Notes in Mathematics 1717. Springer, Berlin.Google Scholar
Blumenthal, R. M. (1992). Excursions of Markov Processes. Birkhauser, Basel.CrossRefGoogle Scholar
Borodin, A. and Salminen, P. (2002). Handbook of Brownian Motion: Facts and Formulae, 2nd edn. Birkhauser, Basel.CrossRefGoogle Scholar
Chen, Z. Q. and Zhao, Z. (1995). Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math. Ann. 302, 323–357.CrossRefGoogle Scholar
Chesney, M., Jeanblanc-Picque, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165–184.10.2307/1427865CrossRefGoogle Scholar
Doob, J. (1957). Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458.10.24033/bsmf.1494CrossRefGoogle Scholar
Forman, J. and Sorensen, M. (2008). The Pearson diffusions: a class of statistically tractable diffusion processes. Scand. J. Statist. 35, 438–465.10.1111/j.1467-9469.2007.00592.xCrossRefGoogle Scholar
Freidlin, M. (1985). Functional Integration and Partial Differential Equations, Annals of Mathematics Studies 109. Princeton University Press.Google Scholar
Graversen, S. and Shiryaev, A. (2000). An extension of P. Levy’s distributional properties to the case of a Brownian motion with drift. Bernoulli 6, 615–620.10.2307/3318509CrossRefGoogle Scholar
Itô, K. (1970). Poisson point processes attached to Markov processes. In Proc. 6th Berkeley Symp. Mathematical Statistics Probability, Vol. 3, pp. 225–239.Google Scholar
Itô, K. and McKean, H. P. (1995). Diffusion Processes and Their Sample Paths. Springer, Berlin.Google Scholar
Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer, Berlin.Google Scholar
Peskir, G. (2006). On reflecting Brownian motion with drift. In Proc. Symp. Stochastic Systems, Osaka 2005, pp. 1–5.10.5687/sss.2006.1CrossRefGoogle Scholar
Peskir, G. (2014). A probabilistic solution to the Stroock–Williams equation. Ann. Prob. 42, 2197–2206.10.1214/13-AOP865CrossRefGoogle Scholar
Pitman, J. and Rogers, L. C. G. (1981). Markov functions. Ann. Prob. 4, 573–582.Google Scholar
Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.CrossRefGoogle Scholar
Pitman, J. and Yor, M. (1997). On the lengths of excursions of some Markov processes. In Seminaire de Probabilités XXXI, eds. J. Azma, M. Emery, M. Ledoux, and M. Yor, Lecture Notes in Mathematics 1655. Springer, Berlin, pp. 272–286.Google Scholar
Pitman, J. and Yor, M. (1999). Laplace transforms related to excursions of a one-dimensional diffusion. Bernoulli 5, 249–255.CrossRefGoogle Scholar
Pitman, J. and Yor, M. (2007). Itô’s excursion theory and its applications. Japanese J. Math. 2, 83–96.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2005). Continous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.Google Scholar
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2. John Wiley, Chichester.Google Scholar
Salminen, P., Vallois, P. and Yor, M. (2007). On excursion theory of linear diffusions. Japanese J. Math. 2, 97–127.CrossRefGoogle Scholar
Zhang, T. (2011). A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients. Ann. Prob. 39, 1502–1527.CrossRefGoogle Scholar
Zhurov, A. I., Levitin, A. L. and Polyanin, D. A. (2020). EqWorld: The World of Mathematical Equations. Available at http://eqworld.ipmnet.ru/index.htm.Google Scholar