8,450 research outputs found
Distributional properties of exponential functionals of Levy processes
We study the distribution of the exponential functional
I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) \d \eta_t, where and
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
is a Brownian motion with drift we show that this integral equation leads to an
important functional equation for the Mellin transform of , which
proves to be a very useful tool for studying the distributional properties of
this random variable. For general L\'evy process ( being Brownian
motion with drift) we prove that the exponential functional has a smooth
density on , but surprisingly the second derivative at zero
may fail to exist. Under the additional assumption that has some positive
exponential moments we establish an asymptotic behaviour of \p(I(\xi,\eta)>x)
as , and under similar assumptions on the negative exponential
moments of we obtain a precise asympotic expansion of the density of
as . Under further assumptions on the L\'evy process
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 has hyper-exponential jumps.Comment: In this version we added a remark after Theorem 1 about extra
conditions required for validity of equation (2.3
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
In this paper we perform the parameter-dependent center manifold reduction
near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf
bifurcations in delay differential equations (DDEs). This allows us to
initialize the continuation of codimension one equilibria and cycle
bifurcations emanating from these codimension two bifurcation points. The
normal form coefficients are derived in the functional analytic perturbation
framework for dual semigroups (sun-star calculus) using a normalization
technique based on the Fredholm alternative. The obtained expressions give
explicit formulas which have been implemented in the freely available numerical
software package DDE-BifTool. While our theoretical results are proven to apply
more generally, the software implementation and examples focus on DDEs with
finitely many discrete delays. Together with the continuation capabilities of
DDE-BifTool, this provides a powerful tool to study the dynamics near
equilibria of such DDEs. The effectiveness is demonstrated on various models
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