Function-measure kernels, self-integrability and uniquely-defined stochastic integrals

Abstract

In this work we study the self-integral of a function-measure kernel and its importance on stochastic integration. A continuous-function measure kernel K over D Ă R d is a function of two variables which acts as a continuous function in the first variable and as a real Radon measure in the second. Some analytical properties of such kernels are studied, particularly in the case of cross-positive-definite type kernels. The self-integral of K over a bounded set D is the "integral of K with respect to itself". It is defined using Riemann sums and denoted ş D Kpx, dxq. Some examples where such notion is well-defined are presented. This concept turns out to be crucial for uniquedefiniteness of stochastic integrals, that is, when the integral is independent of the way of approaching it. If K is the cross-covariance kernel between a mean-square continuous stochastic process Z and a random measure with measure covariance structure M , ş D Kpx, dxq is the expectation of the stochastic integral ş D ZdM when both are uniquely-defined. It is also proven that when Z and M are jointly Gaussian, self-integrability properties on K are necessary and sufficient to guarantee the unique-definiteness of ş D ZdM . Results on integrations over subsets, as well as potential σ-additive structures are obtained. Three applications of these results are proposed, involving tensor products of Gaussian random measures, the study of a uniquely-defined stochastic integral with respect to fractional Brownian motion with Hurst index H ą 1 2 , and the non-uniquely-defined stochastic integrals with respect to orthogonal random measures. The studied stochastic integrals are defined with great generality on the integrand and the integrator and without use of martingale-type conditions, providing a potential filtration-free approach to stochastic calculus grounded on covariance structures.