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The escape probability for integrated Brownian motion with non-zero drift

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

R. A. Atkinson  and
Peter Clifford
R. A. Atkinson*
Affiliation:
University of Birmingham
Peter Clifford*
Affiliation:
University of Oxford
*
∗Postal address: School of Mathematics and Statistics, University of Birmingham, P.O. Box 363, Birmingham, B15 2TT, UK.
∗∗Postal address: Statistics Department, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK.
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Abstract

We consider the two-dimensional process {X(t), V(t)} where {V(t)} is Brownian motion with drift, and {X(t)} is its integral. In this note we derive the joint density function of T and V(T) where T is the time at which the process {X(t)} first returns to its initial value. A series expansion of the marginal density of T is given in the zero-drift case. When V(0) and the drift are both positive there is a positive probability that {Χ (t)} never returns to its initial value. We show how this probability grows for small drift. Finally, using the Kontorovich–Lebedev transform pair we obtain the escape probability explicitly for arbitrary values of the drift parameter.

Keywords

INTEGRATED DIFFUSIONFIRST-PASSAGE TIMEKONTOROVICH–LEBEDEV TRANSFORM

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G17: Sample path properties 60G15: Gaussian processes
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 921 - 929
Copyright
Copyright © Applied Probability Trust 1994 

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