Wiener-Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes

un 2 00 9 A note on Wiener - Hopf factorization for Markov Additive processes

We prove the Wiener-Hopf factorization for Markov Additive processes. We derive also Spitzer-Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.

Lévy processes, the Wiener-Hopf factorisation and applications: part III

1 The M/G/1 Queue Recall that an M/G/1 queue consists of a single server who receives customers at the times of a Poisson process at rate λ > 0 that wait in line to be served on a sequential basis in order of arrival. The i-th customer comes with a workload for the server given by the random variable ξi. The quantities {ξi : i ≥ 1} are i.i.d. The server processes incoming work at a constant (un...

Lévy processes, the Wiener-Hopf factorisation and applications: part I

Let us begin by recalling the definition of two familiar processes, a Brownian motion and a Poisson process. A real-valued process B = {B t : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a Brownian motion if the following hold: (i) The paths of B are P-almost surely continuous. (ii) P(B 0 = 0) = 1. (iii) For 0 ≤ s ≤ t, B t − B s is equal in distribution to B t−s. (iv) For 0 ≤ ...

Lévy processes, the Wiener-Hopf factorisation and applications: part II

On Wiener-Hopf factors for stable processes

We give a series representation of the logarithm of the bivariate Laplace exponent κ of α-stable processes for almost all α ∈ (0, 2].