On the First-Passage Area of a One-Dimensional Jump-Diffusion Process

For a one-dimensional jump-diffusion process X(t), starting from x > 0, it is studied the probability distribution of the area A(x) swept out by X(t) till its first-passage time below zero. In particular, it is shown that the Laplace transform and the moments of A(x) are solutions to certain partial differential-difference equations with outer conditions. The distribution of the maximum displacement of X(t) is also studied. Finally, some explicit examples are reported, regarding diffusions with and without jumps.