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 )) dB t , X (0) = x 0 , through b + Y ( t ), where b > x 0 and Y ( t ) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B t . In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X ( t ) = μt + B t , 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.