Abstract

The aim of this paper is to study the laws of the exponential functionals of the processes X with independent increments , namely I t = t 0 exp(−X s)ds, t ≥ 0, and also I ∞ = ∞ 0 exp(−X s)ds. Under suitable conditions we derive the integro-differential equations for the density of I t and I ∞. We give sufficient conditions for the existence of smooth density of the laws of these function-als. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.

Highlights

  • This study was inspired by the questions arising in mathematical finance, namely by the questions related to perpetuities containing the liabilities, perpetuities subjected to the influence of economical factors

  • The study of exponential functionals is important in the insurance, since the distributions of these functionals appear very naturally in the ruin problem

  • In mathematical finance exponential functionals of the processes with independent increments (PII in short; in what follows, this abbreviation will denote the property of being a process with independent increments) arise very often1

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Summary

Introduction

This study was inspired by the questions arising in mathematical finance, namely by the questions related to perpetuities containing the liabilities, perpetuities subjected to the influence of economical factors (see, for example, Kardaras, Robertson [23]),. In mathematical finance exponential functionals of the processes with independent increments (PII in short; in what follows, this abbreviation will denote the property of being a process with independent increments) arise very often1 This fact is related to the observation that log price is usually not a homogeneous process on a relatively long time interval. Xt = gs−dLs where L is a Lévy process and g is a càdlàg random process independent of L for which the integral is well defined In this case, the conditioned process given σ algebra generated by g, is a PII. Part 2 is devoted to the Kolmogorov type equation for the law of It. It is known that the exponential functional (It )t>0 is not a Markov process with respect to the filtration generated by the process X. In the particular case of I∞ and integrable jumps the equations coincide with the known ones from [15]

Kolmogorov type equation for the density of the law of It
Some results about the existence of smooth densities
When X is a Lévy process

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