Function-measure kernels, self-integrability and uniquely-defined stochastic integrals
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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 ⊂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 ∫_DK(x,dx). Some examples where such notion is well-defined are presented. This concept turns out to be crucial for unique-definiteness 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, ∫_DK(x,dx) is the expectation of the stochastic integral ∫_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 ∫_DZdM. 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 > 1/2, and the non-uniquely-defined stochastic integrals with respect to orthogonal random measures. The studied stochastic integrals are defined without use of martingale-type conditions, providing a potential filtration-free approach to stochastic calculus grounded on covariance structures.
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