This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to −∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions.

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let {B H z , z ∈ [0, 1] 2 } be a fractional Brownian sheet with Hurst parameters H = (H 1 , H 2 ), and ([0, 1] 2 , B([0, 1] 2 ), μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in [0, 1] 2 , and four types of stochastic surface integrals: ϕ(s) dB γ i (s), i = 1, 2, α(a) dB H a , β(a, b) dB H a dB H b , β(a, b) dμ(a) dB H b , β(a, b) dB H a dμ (b). As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ (1/4, 1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.

This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions.