On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process

It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ(X(t)) dBt, X(0) = x0, through b + Y(t), where b > x0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion Bt. In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + Bt , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.