Suprema of Levy processes

Details Suprema of Levy processes Suprema of Levy processes

2011-03-04

arxiv.org/abs/1103.0935v1

Mathematics

Probability

29 pages

Scientific paper

In this paper we study the supremum functional M_t = sup_{s < t} X_s, where X_t is a one-dimensional Levy process. Under very mild assumptions we provide a simple uniform estimate of the cumulative distribution function of M_t. In the symmetric case we find an integral representation of the Laplace transform of the distribution of M_t if the Levy-Khintchin exponent of the process increases on (0,infinity). If the process X_t is a subordinated Brownian motion and the Laplace exponent of the underlying subordinator is a complete Bernstein function, satisfying certain conditions, we are able to find integral formulas for P(M_t < x), as well as of its derivatives in t. Applying these integral formulas we examine the asymptotic behaviour of P(M_t < x) and its t-derivatives, either if t tends to infinity or x tends to 0.

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