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Vector-Valued Stochastic Integration With Respect to Semimartingales in the Dual of Nuclear Space

C. A. Fonseca-Mora

TL;DR

The paper develops a robust theory of vector-valued stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. By employing a regularization approach anchored in real-valued stochastic integration, it defines a stochastic integral that maps into $\Psi'$ for integrands $R$ in $b\mathcal{P}(\Phi',\Psi')$, and proves that the integral is linear, continuous, and compatible with stopping times and continuous parts; it also establishes a Riemann representation and provides criteria for the integral to be a $S^{0}$-good integrator, including the bounded approximation property. The framework enables a well-behaved Itô calculus in spaces of distributions, culminating in a distribution-space Itô formula (extending Üstünel’s results) for distribution-valued semimartingales, and lays the groundwork for nonlinear SPDEs driven by general semimartingale noise in duals of nuclear spaces. These results unify and extend existing real-valued and Hilbert-space stochastic integration theories to the nuclear-space dual setting, facilitating analysis of SPDEs in distribution spaces and tempered distributions. The methods provide concrete tools for approximations (Riemann sums) and enable applications to a broad class of distribution-valued noises, including Lévy-type processes, with implications for stochastic PDE solvers in generalized function spaces.

Abstract

In this work, we investigate a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. Our construction of this particular stochastic integral relies on previous results from [Electron. J. Probab., Volume 26, paper no. 147, 2021], together with specific tools which share some common features with good integrators in finite dimensions. We investigate various properties of this stochastic integral together with applications. In particular we obtain approximations by Riemann sums results, and provide an alternative proof of Üstünel's version of Itô's formula involving of distributions.

Vector-Valued Stochastic Integration With Respect to Semimartingales in the Dual of Nuclear Space

TL;DR

The paper develops a robust theory of vector-valued stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. By employing a regularization approach anchored in real-valued stochastic integration, it defines a stochastic integral that maps into for integrands in , and proves that the integral is linear, continuous, and compatible with stopping times and continuous parts; it also establishes a Riemann representation and provides criteria for the integral to be a -good integrator, including the bounded approximation property. The framework enables a well-behaved Itô calculus in spaces of distributions, culminating in a distribution-space Itô formula (extending Üstünel’s results) for distribution-valued semimartingales, and lays the groundwork for nonlinear SPDEs driven by general semimartingale noise in duals of nuclear spaces. These results unify and extend existing real-valued and Hilbert-space stochastic integration theories to the nuclear-space dual setting, facilitating analysis of SPDEs in distribution spaces and tempered distributions. The methods provide concrete tools for approximations (Riemann sums) and enable applications to a broad class of distribution-valued noises, including Lévy-type processes, with implications for stochastic PDE solvers in generalized function spaces.

Abstract

In this work, we investigate a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. Our construction of this particular stochastic integral relies on previous results from [Electron. J. Probab., Volume 26, paper no. 147, 2021], together with specific tools which share some common features with good integrators in finite dimensions. We investigate various properties of this stochastic integral together with applications. In particular we obtain approximations by Riemann sums results, and provide an alternative proof of Üstünel's version of Itô's formula involving of distributions.

Paper Structure

This paper contains 13 sections, 26 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nuclear spaces and their strong duals
  4. Cylindrical and Stochastic Processes
  5. Real-valued stochastic integration
  6. $S^{0}$-good integrators
  7. Vector-Valued $S^{0}$-Stochastic Integration
  8. Construction of the vector-valued $S^{0}$-stochastic integral
  9. Properties of the $S^{0}$-stochastic integral
  10. Extension of the $S^{0}$-stochastic integral
  11. Riemann Representation
  12. The integral as a $S^{0}$-good integrator
  13. Itô's formula in the space of distributions

Key Result

Theorem 3.2

Assume $\Phi$ is either a Fréchet nuclear space or the strict inductive limit of Fréchet nuclear spaces. Let $X$ be a $(\mathcal{F}_{t})$-adapted semimartingale in $\Phi'$ for which the mapping $X:\Phi \rightarrow S^{0}$ is continuous from $\Phi$ into $S^{0}$. The following assertions are equivalent

Theorems & Definitions (68)

  • Remark 2.1
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Corollary 3.6
  • ...and 58 more