Tail asymptotics for exponential functionals of Lévy processes

Abstract

Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional Z ¼ R 1 0 e ÀX ðtÞ dt of a Le´vy process X ðtÞ; tX0. In particular, we investigate its tail asymptotics. We show that, depending on the right tail of X ð1Þ, the tail behavior of Z is exponential, Pareto, or extremely heavy-tailed. r

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

“…Again we require a result, given in Section 3.3, for a fairly general class of processes with independent heavy-tailed increments. The specialisation of this result, under appropriate conditions, to an (unmodulated) Lévy process gives a simple proof of the continuous-time version of the Pakes-Veraverbeke Theorem, different from that found in the existing literature-see, e.g., Klüppelberg, Kyprianou and Maller (2004) and Maulik and Zwart (2005).…”

“…Detailed proofs may be found in Foss and Zachary (2002). We prove (27) under the assumption that the regenerative process X is aperiodic, so that the distance ||π n − π|| between π n and π in the total variation norm tends to zero-the modifications required to deal with the periodic case are routine. Then…”

“…As in the Pakes-Veraverbeke Theorem for unmodulated random walks, the intuitive idea underlying the above result is the following: the maximum M will exceed a large value y if the process follows the typical behaviour specified by the law of large numbers, i.e. it's mean path, except that at any time n a jump occurs of size greater than y + na; this has probability E[F Xn (y + na)], and so the bound is now given by the use of (27).…”

Some Time-Dependent Properties of Symmetric M/G/1 Queues

“…Next, observe that E[e u * Y ] = λ/(λ − u * ) and Ee u * X = β λ (−u * ) are finite. Applying again [15] we get…”

On a Theorem of Breiman and a Class of Random Difference Equations

“…Unfortunately, not many results are available when Q is light tailed and |M| ≤ 1. Some partial results can be found in [11] and [19]. A relatively clean case seems to be when e Q is regularly varying with index −α and independent of M. After taking exponents in (1.2), we may wonder whether the application of Breiman's theorem is justified, i.e.…”