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Explicit asymptotics on first passage times of diffusion processes

Part of: Mathematical finance Distribution theory Stochastic processes

Published online by Cambridge University Press:  15 July 2020

Angelos Dassios  and
Luting Li
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Luting Li*
Affiliation:
London School of Economics and Political Science
*
*Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email addresses: [email protected], [email protected]
*Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email addresses: [email protected], [email protected]
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Abstract

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

Keywords

First passage timediffusion processperturbation theoryOrnstein–Uhlenbeck processBessel processexponential-Shiryaev processhypergeometric diffusionspecial functions

MSC classification

Primary: 91G60: Numerical methods (including Monte Carlo methods)
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62E17: Approximations to distributions (nonasymptotic) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 681 - 704
Copyright
© Applied Probability Trust 2020

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