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：本文研究了函数-测度核的自积分及其在随机积分中的重要性。定义在D ⊂ ℝ^d上的连续函数-测度核K是一个二元函数，它在第一个变量上表现为连续函数，在第二个变量上表现为实Radon测度。本文研究了此类核的一些解析性质，特别是交叉正定核的情形。K在有界集D上的自积分是“K关于自身的积分”，通过Riemann和定义，记为∫_D K(x,dx)。文中给出了该概念良定义的一些例子。这一概念对于随机积分的唯一确定性至关重要，即积分与逼近方式无关。若K是均方连续随机过程Z与具有测度协方差结构M的随机测度之间的交叉协方差核，则当两者均唯一确定时，∫_D K(x,dx)即为随机积分∫_D ZdM的期望。本文还证明，当Z与M联合高斯时，K的自可积性是保证∫_D ZdM唯一确定的充要条件。此外，还得到了子集上的积分结果以及潜在的σ-可加结构。本文提出了这些结果的三项应用，涉及高斯随机测度的张量积、关于Hurst指数H > 1/2的分数布朗运动的唯一确定随机积分的研究，以及关于正交随机测度的非唯一确定随机积分。所研究的随机积分无需使用鞅型条件定义，为基于协方差结构的随机微积分提供了一种潜在的无滤波方法。