 
Summary: STOCHASTIC INTEGRALS AND ABELIAN PROCESSES
CLAUDIO ALBANESE
Abstract. We study triangulation schemes for the joint kernel of a diffusion process with
uniformly continuous coefficients and an adapted, nonresonant Abelian process. The proto
typical example of Abelian process to which our methods apply is given by stochastic integrals
with uniformly continuous coefficients. The range of applicability includes also a broader class
of processes of practical relevance, such as the sup process and certain discrete time summa
tions we discuss.
We discretize the space coordinate in uniform steps and assume that time is either con
tinuous 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 FeynmanKac,
Girsanov, Ito and CameronMartin, which are also reobtained.
We make use of a pathwise analysis without relying on a probabilistic interpretation. The
Fourier representation is needed to regularize the hypoelliptic character of the joint process
of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier
