Joint Distribution of First-Passage Time and First-Passage Area of Certain Levy Processes

Let be X(t) = x - mu t + sigma B-t - N-t a Levy process starting from x > 0, where mu >= 0, sigma >= 0, B-t is a standard BM, and N-t is a homogeneous Poisson process with intensity > 0, starting from zero. We study the joint distribution of the first-passage time below zero, tau(x), and the first-passage area, A(x), swept out by X till the time tau(x). In particular, we establish differential-difference equations with outer conditions for the Laplace transforms of tau(x) and A(x), and for their joint moments. In a special case (mu = sigma = 0), we show an algorithm to find recursively the moments E[tau(x)(m)A(x)(n)], for any integers m and n; moreover, we obtain the expected value of the time average of X till the time tau(x).