Distributional properties of exponential functionals of Levy processes

Abstract

We study the distribution of the exponential functional $I(ξ,η)=\int_0^{\infty} \exp(ξ_{t-}) \d η_t$, where $\xi$ and $\eta$ are independent Lévy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $\eta$ is a Brownian motion with drift we show that this integral equation leads to an important functional equation for the Mellin transform of $I(\xi ,\eta )$, which proves to be a very useful tool for studying the distributional properties of this random variable. For general Lévy process $\xi$ ($\eta$ being Brownian motion with drift) we prove that the exponential functional has a smooth density on $\mathring{\setminus }\{0\}$, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that $\xi$ has some positive exponential moments we establish an asymptotic behaviour of $\p(I(ξ,η)>x)$ as $x\to +\infty$, and under similar assumptions on the negative exponential moments of $\xi$ we obtain a precise asympotic expansion of the density of $I(\xi ,\eta )$ as $x\to 0$. Under further assumptions on the Lévy process $\xi$ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when $\xi$ has hyper-exponential jumps.

In this version we added a remark after Theorem 1 about extra conditions required for validity of equation (2.3)

References in corpus (4)

• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• On the distribution of exponential functionals for Levy processes with jumps of rational transform
• On the density of exponential functionals of Lévy processes
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Cited by in corpus (7)

• A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes
• On the law of killed exponential functionals
• Continuity properties and the support of killed exponential functionals
• Exponential functionals of Lévy processes with jumps
• Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions
• Statistical inference for exponential functionals of Lévy processes
• On exponential functionals of Levy processes