A Hull and White formula for a stochastic volatility Lévy model with infinite activity

Abstract

The new class of Gaussian processes which are obtained by a compact perturbation in the reproducing kernel of Wiener process is in- troduced. The nite families of increments of our processes on small time intervals behave as increments of Wiener process. Consequently a lot of asymptotical properties of Wiener process are inherited. The law of iterated logarithm, the analogue of the Levy modulus of continuity and almost unifor- mity of hitting distribution on the small circles are proved. The renormalized Fourier{Wiener transform of the self-intersection local time for compactly perturbed Wiener process is constructed. In this article we consider self-intersection local times for one class of planar Gaussian processes. The interest to the self-intersection local times of planar Brownian motion has a long history since the theorem of Dvoretzky, Erdos, Kaku- tani (3) which established the existence of multiple self-intersections. The various kinds of renormalization were proposed for the self-intersection local times of pla- nar Brownian motion and certain Levy processes in the articles (4, 5, 10, 15, 16). Most of these articles essentially use the Markov property of the considered pro- cess. The aim of this paper is to present an approach to investigation of planar Gaussian processes which does not use the Markov property. We introduce a new class of Gaussian processes which are obtained with the help of compact pertur- bation in the reproducing kernel of Wiener process. Such processes inherit many properties of Wiener process. As an example we prove here the law of iterated logarithm, the analogue of the Levy modulus of continuity and almost uniformity of hitting distribution on the small circles. The nite families of small increments of our processes behave like the increments of Wiener process. This allows us to prove that compactly perturbed Wiener process has strong local nondeterminism property, which is a generalization of local nondeterminism property introduced by S.Berman (1). Finally we present the renormalization for Fourier{Wiener trans- form of the self-intersection local times for compactly perturbed Wiener process.