On exponential functionals of Lévy processes
نویسندگان
چکیده
Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate properties of the mapping Φ, which maps two independent Lévy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective and give the inverse mapping in terms of (Lévy) characteristics. Also, continuity of Φ is treated and some results on its range are obtained. 2000 Mathematics subject classification. 60G10, 60G51, 60J35.
منابع مشابه
Exponential Functionals of Lévy Processes
This text surveys properties and applications of the exponential functional R t 0 exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0).
متن کاملTail asymptotics for exponential functionals of Lévy processes
Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional Z = ∫∞ 0 e−X(t)dt of a Lévy process X(t), t ≥ 0. In particular, we investigate its tail asymptotics. It is shown that, depending on the right tail of X(1), the tail behavior of Z is exponential, Pareto, or extremely heavy-tailed.
متن کاملApplying the Wiener-Hopf Monte Carlo Simulation Technique for Lévy processes to Path Functionals such as First Passage Times, Undershoots and Overshoots
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Lévy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Lévy model) and insurance...
متن کاملInfinitely Divisibility of Solutions of Some Semi-stable Integro-differential Equations and Exponential Functionals of Lévy Processes
0 (1 ∧ x2) ν(dx) < +∞, are uniquely determined by the distribution of a spectrally negative infinitely divisible random variable, with characteristic exponent ψ. L(α,ψ,γ) is known to be the infinitesimal generator of a 1 α -semi-stable Feller semigroup on R+, which has been introduced by Lamperti [18]. The functions are expressed in terms of a new family of power series which includes, for inst...
متن کاملAdditive functionals of several Lévy processes and self-intersection local times
Different extentons of an isomorphism theorem of Dynkin are developed and are used to study two distinct but related families of functionals of Lévy processes; n-fold “near-intersections” of a single Lévy process, which is also referred to as a self-intersection local time, and continuous additive functionals of several independent Lévy processes. Intersection local times for n independent Lévy...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012