On First Passage Times and Differential Equations,

Abstract

Abstract : Practical and theoretical considerations in computing first passage time statistics are considered. We are motivated by first passage times as models of failure times. In Section 1, the relevance of first passage time distrbutions as failure time models is indicated. Also, the spectral series expansion solution to the backward equation is introduced. In Section 2, algorithms for approximating w(x,t) are obtained. In particular, the infinite spectral expansion of rw(x,t) is approximated by an n-term sub-expansion which matches the first n-1 moments. Proofs validating the spectral expansion and the related approximation scheme are given in the Appendix. In Sections 3 and 4, methods are given for obtaining the eigenvalues and first passage moments, necessary for computing approximations to w(x,t). In Section 5, computational issues related to calculating the moment generating function are considered. Sections 6 and 7 include theoretical complements about first passage times. In particular, the moment generating function is shown to possess an interesting representation having exponential form (cf equations (7.1)). This exponential representation is related to asymptotic expansions used in analyzing perturbations of certain second-order differential equations.