Related papers: Finite Variation of Fractional Levy Processes

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention in particular the many cases of spectrally negative…

Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and…

We derive a necessary and sufficient condition for stochastic processes to have almost periodic finite dimensional distributions; in particular, we obtain characterizations for infinitely divisible processes to be almost periodic in terms…

It is shown that a certain functional of a branching process has representations in terms of both a maximisation problem and a minimisation problem. A consequence of these representation is that upper and lower bounds on the functional can…

We study such important properties of $f$-divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for $f$-divergence…

Levy flights and subdiffusive processes and their properties are discussed. We derive the space- and time-fractional transport equations, and consider their solutions in external potentials. An extensive list of references is included.

We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive L\'evy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction…

Using generalized Blumenthal--Getoor indices, we obtain criteria for the finiteness of the $p$-variation of L\'evy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations…

We discuss the Gamma Levy process, including path properties, the inverse process, integrability, and its spin-offs obtained by compounding, exponentiation, and other operations; further extendable to arbitrary sigma-finite continuous Borel…

Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler-Lagrange equations are derived. Fractional equations of motion are…

We study the scale function of the spectrally negative phase-type Levy process. Its scale function admits an analytical expression and so do a number of its fluctuation identities. Motivated by the fact that the class of phase-type…

The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Levy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also…

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example,…

In this work we give a complete description to the asymptotic behaviors of exponential functionals of L\'evy processes and divide them into five different types according to their convergence rates. Not only their exact convergence speeds…

This paper deals with the large deviations behavior of a stochastic process called thinned Levy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs. The process has a…

We construct the law of L\'{e}vy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of…

In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\alpha$ of a continuous local martingale, where $\alpha\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order…

We consider one-parameter families of quadratic-phase integral transforms which generalize the fractional Fourier transform. Under suitable regularity assumptions, we characterize the one-parameter groups formed by such transforms.…

To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law…