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Distributional properties of exponential functionals of Levy processes | | A. Kuznetsov
; J.C. Pardo
; M. Savov
; | | Date: |
31 May 2011 | | Abstract: | We study the distribution of the exponential functional
$I(xi,eta)=int_0^{infty} exp(xi_{t-}) d eta_t$, where $xi$ and $eta$
are independent L’evy 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 L’evy process $xi$ ($eta$ being Brownian
motion with drift) we prove that the exponential functional has a smooth
density on $
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 $p(I(xi,eta)>x)$
as $x o +infty$, 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 $x o 0$. Under further assumptions on the L’evy 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. | | Source: | arXiv, 1105.6365 | | Services: | Forum | Review | PDF | Favorites | |
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