First-passage problems for diffusion processes with state-dependent jumps

Abstract

Let X(t) be a time-homogeneous jump-diffusion process. We assume that the jump size depends on the value of X(t). We obtain analytical results for the moments of T(x) and of $X(T(x)),$ where T(x) is the first time that the process leaves the interval (a, b). We also compute $P[X(T(x))\le a].$ These results have applications in financial mathematics.

Keywords:

• First-passage time
• Brownian motion
• Poisson process
• jump size
• differential-difference equation

AMS Subject Classification::

• 60J75
• 60J60

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada.