Integration theory for infinite dimensional volatility modulated Volterra processes

TL;DR

This paper develops a stochastic integration framework for infinite-dimensional, volatility-modulated Volterra processes using Malliavin calculus, including an Itô formula and applications to PDE solutions.

Contribution

It introduces a new stochastic integration theory for Hilbert-valued Volterra processes with volatility modulation, combining Malliavin calculus and random-field approaches.

Findings

Defined a stochastic integral with properties similar to classical ones

Derived an Itô formula for the new integral

Connected the integration theory to PDE fundamental solutions

Abstract

We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an It\^{o} formula. Moreover, we derive an alternative definition using a random-field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.