Investigating the distribution of the first-crossing area of a diffusion process with jumps over a threshold

For a given barrier S and a one-dimensional jump-diffusion process X (t), starting from x < S, we study the probability distribution of the integral (Formula presented.) determined by X (t) till its first-crossing time (Formula presented.) over S. In particular, we show that the Laplace transform and the moments of A (x) S are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of X (t) in [0, πs(x)] is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by X (t) till its firstpassage below zero. Some explicit examples are reported, regarding diffusions with and without jumps.