Efficient estimation of first passage time density function for jump-diffusion processes

• Published in: SIAM journal on scientific computing Vol. 26; no. 5; pp. 1760 - 1775
• Main Authors: ATIYA, Amir F, METWALLY, Steve A. K
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2005
• Subjects: Algorithms; Applied mathematics; Bias; Brownian motion; Exact sciences and technology; Mathematical models; Mathematics; Methods; Monte Carlo simulation; Neurosciences; Nonparametric inference; Normal distribution; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Statistics; Stochastic processes; Monte Carlo method; 60G20; Finance; inverse Gaussian density 62G07; Probability distribution; Numerical method; Stochastic method; Stochastic process; Poisson process; Inverse Gaussian density; Brownian bridge; Brownian motion; Probability density; Numerical analysis; Jump diffusion; Diffusion process; First passage time; Jump process; 65C05; Computing method; Analytical solution; Monte Carlo simulation; jump-diffusion
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/s1064827502417982

The first passage time problem has attracted considerable research interest in the field of stochastic processes. It concerns the estimation of the probability density of the time for a random process to cross a specified boundary level. Even though there are many theoretical advances in solving this problem, for many classes of random processes no analytical solution exists. The jump-diffusion process (JDP) is one such class. Recent research in finance theory has renewed the interest in JDPs, and the first passage time problem for such processes is applicable to several finance problems. In this paper we develop fast Monte Carlo-type numerical methods for computing the first passage density function for JDPs. Compared with the standard Monte Carlo approach, our approaches are about 10--30 times faster.