On the Distributions of the Sup and Inf of the Classical Risk Process with Exponential Claim*

Here x0 ≥ 0 is the initial capital, c > 0 is the premium income per unit of time, N = {Nt, t ≥ 0} is an homogeneous Poisson process with rate λ and {Rk, k = 1, 2, . . .} is a sequence of i.i.d. random variables independent ofN . Henceforth we suppose that R1 has exponential distribution with mean 1/r. Our goal is to calculate explicit expressions for the distributions of the random variables sup{Xs : s ≤ t} and inf{Xs : s ≤ t}, t > 0. Toward this end, we apply the complex inversion theorem of the Laplace transform (or Lerch’s theorem) to the double Laplace transforms of some occupation measures of X (see Section 2 below). As a consequence, we are also able to give the distribution of the first passage of certain level x ∈ R of the process X . It means, the distributions of Sx = inf{t > 0 : Xt = x} and Tx = inf{t > 0 : Xt < x},