Title: Vector-Valued Stochastic Integration With Respect to Semimartingales in the Dual of Nuclear Space Show Chinese title

Title: 向量值随机积分关于核空间对偶中的半鞅

Abstract: In this work, we introduce a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. These semimartingales are required to have the good integrator property, which is a property that we explore in detail and provide several examples. Our construction of the stochastic integral uses a regularization argument for cylindrical semimartingales and the theory of real-valued stochastic integration introduced by the author in a previous work [Electron. J. Probab., Volume 26, paper no. 147, 2021]. We show various properties of the stochastic integral; in particular we study continuity of the integral mapping for integrands and for integrators, we prove a Riemman representation formula, and we introduce sufficient conditions for the stochastic integral to be a good integrator. Finally, we apply our theory to show an extension of \"{U}st\"{u}nel's version of It\^{o}'s formula in the spaces of distributions and of tempered distributions. Show Chinese abstract

Abstract: 在本工作中，我们引入了关于取值于核空间对偶的半鞅的算子值过程的随机积分理论。 这些半鞅需要具有良好的积分器性质，这是一个我们详细探讨并提供多个例子的性质。 我们构造随机积分使用了圆柱半鞅的正则化论证以及作者在之前的工作中引入的实值随机积分理论 [Electron. J. Probab., 第26卷，论文编号147，2021年]。 我们展示了随机积分的各种性质；特别是我们研究了积分映射对于被积函数和积分器的连续性，证明了一个Riemman表示公式，并为随机积分成为良好积分器提供了充分条件。 最后，我们将我们的理论应用于展示Üstünel的分布空间和缓增分布空间中的Itô公式的扩展。