Stochastic Integrals and Abelian Processes

Abstract

We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coeffcients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summations we discuss. We discretize the space coordinate in uniform steps and assume that time is either continuous or finely discretized as in a fully explicit Euler method and the Courant condition is satisfied. We show that the Fourier transform of the joint kernel of a diffusion and a stochastic integral converges in a uniform graph norm associated to the Markov generator. Convergence also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic integrals are straightforward to define for finite triangulations and the convergence result gives a new and entirely constructive way of defining stochastic integrals in the continuum. The method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac, Girsanov, Ito and Cameron-Martin, which are also re-obtained. We make use of a path-wise analysis without relying on a probabilistic interpretation. The Fourier representation is needed to regularize the hypo-elliptic character of the joint process of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier analysis framework can be generalized. This condition leads to the notion of non-resonant Abelian process.