Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-04T07:12:33.989Z Has data issue: false hasContentIssue false
  • English
  • Français

Smooth first-passage densities for one-dimensional diffusions

Published online by Cambridge University Press:  14 July 2016

E. J. Pauwels
E. J. Pauwels*
Affiliation:
Limburgs Universitair Centrum
*
∗Postal address: Limburgs Universitair Centrum, B-3610 Diepenbeek, Belgium. Research partially supported by NFWO (Belgium).
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density.

In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

Keywords

BESSEL (3)-PROCESSSMOOTHNESS CONDITIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 2 , June 1987 , pp. 370 - 377
Copyright
Copyright © Applied Probability Trust 1987 

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

Ikeda, N. and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Pitman, J. W. (1975) One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Prob. 7, 511–526.Google Scholar
Rogers, L. C. G. (1985) Smooth transition densities for one-dimensional diffusions. Bull. London Math. Soc. 17, 157–161.CrossRefGoogle Scholar
Rogers, L. C. G. and Pitman, J. W. (1981) Markov functions. Ann. Prob. 9, 573–582.CrossRefGoogle Scholar