First-Passage Problems for One-Dimensional Diffusions with Random Jumps from a Boundary

Let X(t) be a time-homogeneous one-dimensional diffusion process defined in I subset of IR, starting at x is an element of I and let c is an element of I a barrier with c < x. Suppose that, whenever the barrier c is reached, the process X is killed and it is continued as a new process <(X)over tilde> which makes a random jump from c according to a given distribution, and then it starts again. First-passage problems for (X) over tilde are investigated, and closed form expressions are found for the expectation and the moment generating function of the first-passage time of (X) over tilde over a boundary S > x, and for the first-exit time of (X) over tilde from an open interval containing c, when the infinitesimal coefficients are allowed to change as (X) over tilde crosses the boundary c. Moreover, the stationary distribution of (X) over tilde is studied, whenever it exists. Some explicit examples are also reported.