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

Let be X ( t ) = x − μ t + σ B t − N t a Lévy process starting from x > 0, where μ ≥ 0, σ ≥ 0, B t is a standard BM, and N t is a homogeneous Poisson process with intensity θ > 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 ) 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 τ ( x ).