Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

TL;DR

This paper extends infinite dimensional calculus by introducing weak Dirichlet processes, enabling new stochastic PDE analysis and optimal control verification in Hilbert space settings.

Contribution

It introduces the concept of weak Dirichlet processes in infinite dimensions, providing a new decomposition useful for stochastic calculus and control.

Findings

Weak Dirichlet processes decompose into local martingale and orthogonal parts.

The decomposition serves as a substitute for Itô's formula in infinite dimensions.

Application to verification theorem for stochastic optimal control problems.

Abstract

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$, taking values in a Hilbert space $H$, is the sum of a local martingale and a suitable "orthogonal" process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an It\^o type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of "mild solution" for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.