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 $D \subset \mathbb{R}^{d}$ 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 $\int_{D}K(x,dx)$. 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 $M$, $\int_{D}K(x,dx)$ is the expectation of the stochastic integral $\int_{D} ZdM$ 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 $\int_{D}ZdM$. Results on integrations over subsets, as well as potential $\sigma$-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 $H > \frac{1}{2}$, 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.
翻译:在此工作中, 我们研究一个函数测量核心的自我整体性, 以及它对于随机集成的重要性。 一个连续功能测量核心$K$超过$D的自动交易 =subset\ mathbb{R<unk> d}$是两个变量的函数, 它在第一个变量中是一个连续函数, 在第二个变量中是一个真正的拉多度量。 这些内核的某些分析属性, 特别是在跨正向确定型内核的内核内核内核。 与一个约束型内核的美元相比, 美元对一个自定义的内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内 内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内核内</s>