Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion - Approximation, quadratic variation, and It\^o formula

TL;DR

This paper investigates infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion operators, focusing on their approximation, quadratic variation, and an associated Itô formula within the framework of stochastic calculus.

Contribution

It introduces a novel analysis of infinite dimensional OU processes with unbounded diffusion, including convergence of finite-dimensional approximations and new quadratic variation formulas.

Findings

Weak convergence of finite-dimensional processes to the infinite-dimensional process

Explicit calculation of scalar quadratic variation for the infinite-dimensional process

Derivation of an Itô formula for these processes

Abstract

The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron-Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related It\^o formula is presented.