Occupation Times of General Lévy Processes

Abstract

For an arbitrary Lévy process X which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of X and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of X. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications.

On weighted occupation times for refracted spectrally negative Lévy processes

“…With (24) for ω(x) = p + (q − p)1 {x<a} , we here only compare the ratios in Corollary 3 with existing results from [14, Theorem 5 and 6] and [22, Corollary 1 and 2] for c = 0.…”

“…[14] focuses on the existence and uniqueness of solution to (1) and some fluctuation identities for X are also established. [15] investigats the occupation times of half lines, [22] identifies the distribution of various functionals related, and [25,24] mainly consider general Lévy process with rational jumps.…”

“…During the last several years there have been a series of papers concerning occupation time related problems for SNLP. These problems arise from both theoretical interests and the applications in risk theory and finance; see for example [6,16,21,18,19,17,15,22,25,24]. Among them using a perturbation approach, [16] studies the occupation times of semi-infinite intervals.…”

Occupation times of alternating renewal processes with Lévy applications

“…In [15], [33], and [34] occupation times of spectrally negative Lévy processes were studied, while in [32] refracted Lévy processes were dealt with; these results are typically occupation times until a first passage time. Occupation times up to a fixed time horizon t have been studied in [25] for spectrally negative Lévy processes and in [45] for a general Lévy process that is not a compound Poisson process. The cases in which X(•) is a Brownian motion, or Markov-modulated Brownian motion have been extensively studied; see e.g.…”