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

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