Investigating the Distribution of the First-Crossing Area of a Diffusion Process with Jumps Over a Threshold

Mario Abundo
2013 Open Applied Mathematics Journal  
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 A_S(x)= ∫_0 ^ τ_S(x)X(t) dt determined by X(t) till its first-crossing time τ_S(x) over S. In particular, we show that the Laplace transform and the moments of A_S(x) are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of X(t) in [0, τ_S(x)] is also studied. Thus, we extend the
more » ... ts of a previous paper by the author, concerning the area swept out by X(t) till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps.
doi:10.2174/1874114201307010018 fatcat:wbjmykgyongs3e3pi4efwsowxa