Approximating the First Passage Time Density of Diffusion Processes with State-Dependent Jumps

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations on their first passage time probability density functions and results for the qualitative behavior of other statistics of this random variable. Finally, we provide two examples of application of the method for different choices of the distribution involved in the mechanism of generation of the jumps.

Automated summary

We investigate the problem of first passage time for a jump diffusion process with a nonlocal Jacobi operator as its infinitesimal generator. Due to the lack of analytical solutions, we propose a discretization scheme to simulate the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations for the first passage time probability density functions and study the qualitative behavior of other statistics associated with this random variable. Additionally, we provide two examples illustrating the application of this method with different jump generation mechanisms.