First passage problems for one-dimensional diffusions with random jumps from a boundary

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