Boundary Crossing Probabilities of Jump Diffusion Processes to Time-Dependent Boundaries

The finite Markov chain imbedding technique has been used to compute the boundary crossing probabilities of one and higher-dimensional Brownian motion. The idea is to cast the boundary crossing probabilities as the limiting probabilities of a finite Markov chain entering a set of absorbing states induced by the boundaries. In this manuscript, we extend the technique to compute the boundary crossing probabilities of a class of jump diffusion processes to time-dependent boundaries. We allow the jump sizes to have general distributions and the boundaries to be non-linear. Numerical examples are given to illustrate our theoretical results.