Exact solution to a first-passage problem for an Ornstein-Uhlenbeck process with jumps and its integral

Let dX(t) =Y(t) dt, where Y (t) is an Ornstein-Uhlenbeck process with Poissonian jumps, and let T (x,y) be the first time that X(t) + Y(t) = 0, given that X (0) = x and Y (0) = y. The moment-generating function of T (x,y) is obtained in the case when the jumps are exponentially distributed by solving the integro-differential equation it satisfies, subject to the appropriate boundary conditions. The case when the jumps are uniformly distributed is also considered.

Automated summary

This passage describes the analytical solution method for an Ornstein-Uhlenbeck process with Poissonian jumps, considering both exponentially and uniformly distributed jumps.