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Hitting-Time Densities of a Two-Dimensional Markov Process

Published online by Cambridge University Press:  27 July 2009

C. H. Hesse
C. H. Hesse*
Affiliation:
Department of Statistics 367 Evans Hall University of California, Berkeley Berkeley, California 94720
*
Current address: Mathematisches Institut A, Universität Stuttgart, D-7000 Stuttgart 80, Germany.
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Abstract

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

Information

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 4 , October 1992 , pp. 561 - 580
Copyright
Copyright © Cambridge University Press 1992

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