 
Summary: EFFICIENT ESTIMATION OF FIRST PASSAGE TIME DENSITY
FUNCTION FOR JUMP DIFFUSION PROCESSES
AMIR F. ATIYA AND STEVE A.K. METWALLY
Abstract. 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 is one such class. Recent research in finance theory has renewed the interest in jump
diffusion processes, and the first passage time problem for such processes is applicable to several
finance problems. In this paper we develop fast Monte Carlotype numerical methods for computing
the first passage density function for jumpdiffusion processes. Compared with the standard Monte
Carlo approach, our approaches are about 1030 times faster.
keywords: Monte Carlo Simulation, Brownian Motion, Brownian
Bridge, Jump Diffusion, First Passage Time, Poisson process, inverse
Gaussian density.
1. Introduction. Jump diffusion processes have experienced renewed interest
in the theory of finance. It became widely acknowledged in the academic literature
that the traditionally used geometric Brownian motion model for market behavior
falls short of explaining empirical observations of market returns and its underlying
derivatives' prices [4], [24], [6]. Markets exhibit fattailed behavior, and for that
