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 $\xi$ 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 $\r \setminus \{0\}$, 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 $x\to +\infty$, 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 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.Comment: In this version we added a remark after Theorem 1 about extra conditions required for validity of equation (2.3

The extended hypergeometric class of L\'evy processes

With a view to computing fluctuation identities related to stable processes, we review and extend the class of hypergeometric L\'evy processes explored in Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of a process in the extended class, and characterise its exponential functional. Finally, we give three concrete examples arising from transformations of stable processes.Comment: 22 page

Boundary Dynamics Driven Entanglement

We will show how it is possible to generate entangled states out of unentangled ones on a bipartite system by means of dynamical boundary conditions. The auxiliary system is defined by a symmetric but not self-adjoint Hamiltonian and the space of self-adjoint extensions of the bipartite system is studied. It is shown that only a small set of them leads to separable dynamics and they are characterized. Various simple examples illustrating this phenomenon are discussed, in particular we will analyze the hybrid system consisting on a planar quantum rotor and a spin system under a wide class of boundary conditions.Comment: 26 pages, 5 figure

A Wiener--Hopf Monte Carlo simulation technique for L\'{e}vy processes

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general L\'{e}vy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr's so-called "Canadization" technique as well as Doney's method of stochastic bounds for L\'{e}vy processes; see Carr [Rev. Fin. Studies 11 (1998) 597--626] and Doney [Ann. Probab. 32 (2004) 1545-1552]. We rely fundamentally on the Wiener-Hopf decomposition for L\'{e}vy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos L\'{e}vy en titillant la factorization de Wiener-Hopf (2002) Laboratoire de Math\'{e}matiques de L'INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801--1830]. We illustrate our Wiener--Hopf Monte Carlo method on a number of different processes, including a new family of L\'{e}vy processes called hypergeometric L\'{e}vy processes. Moreover, we illustrate the robustness of working with a Wiener--Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given L\'{e}vy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.Comment: Published in at http://dx.doi.org/10.1214/10-AAP746 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org