On the approximation of Lévy driven Volterra processes and their integrals

Abstract

Volterra processes appear in several applications ranging from turbulence to energy finance where they are used in the modelling of e.g. temperatures and wind and the related financial derivatives. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. In this work we suggest to construct an approximating sequence of Lévy driven Volterra processes, by perturbation of the kernel function. In this way, one can obtain an approximating sequence of semimartingales.Then we consider fractional integration with respect to Volterra processes as integrators and we study the corresponding approximations of the fractional integrals. We illustrate the approach presenting the specific study of the Gamma-Volterra processes. Examples and illustrations via simulation are given.

On a Stochastic Linear Dynamical System Driven by a Volterra Process

“…Example. In a work by G.Di Nunno and al [8] on approximation of Lévy -driven Volterra processes, the authors consider a very interesting example on Gamma Volterra proess t 0 (t − s) β e −γ(t−s) dL s . For the kernel of this process h(u) = u β e −γu we can put du = u β+1 e −γu dx.…”

“…In some works on Lévy driven Volterra processes and their applications as in [8], one needs to have an expression for the differential of Y t or U t .…”

“…H is Hurst index, 0 < H < 1 [10,13]. Many things have been done for study of Volterra processes in various types and its applications where driving processes can be some random measure or some stochastic process such as semimartingale while the kernel K(s, t) is some real deterministic function (refer to [1,2,3,8] or [5,6]).…”

“…And for such Volterra processes, in a most recently works by G.D.Nunno and al. [8], a kind of semimartingale approximation has been considered with applications to construction of pathwise fractional Volterra integration.…”

Stochastic differential equations driven by additive Volterra-Lévy and Volterra-Gaussian noises

“…where u : R → R is a measurable function, and Y = {Y t , t ≥ 0} is a Volterra-Lévy process. Equations of the form (1), with different coefficients and different noises, were the subject of long and careful considerations. Namely, the most popular case is the Langevin equation, where u(x) = ax, x ∈ R, with some coefficient a 0, and a Wiener process as a noise.…”