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

Abstract

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.