First passage time distributions for jump-diffusion processes and flexible boundaries

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Abstract: The first passage time (FPT) is a useful tool in stochastic modeling of many biological, physical, social and economic processes evolving with time. It refers to the time when a random process first passes a threshold, e.g., when the population of an endangered species reaches a certain critical level, or when the number of infected individuals with a disease reaches a limit. Other examples include the survival time of a cancer patient, failure time of a mechanical system, and default time of a business, etc.

We study the boundary crossing problem for jump-diffusion processes over a discontinuous boundary and provide a complete characterization on the FPT distributions. We derive new formulas for piecewise linear boundary crossing probabilities and density of Brownian motion with general random jumps. These formulas can be used to approximate the boundary crossing distributions for general nonlinear boundaries. The method can be extended to more general diffusion processes such as geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps. The numerical computation can be done by Monte Carlo integration which is straightforward and easy to implement. Some numerical examples are presented for illustration.