From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration

TL;DR

This paper introduces a unified framework for stochastic integration with respect to Gaussian fields, extending classical integrals to random integrands using chaos decomposition, facilitating analysis and computation.

Contribution

It develops a unified approach to stochastic integration for Gaussian fields, connecting Ito-Skorokhod and Stratonovich integrals through chaos space decomposition.

Findings

Provides an efficient method for analyzing stochastic integrals

Extends integration definitions to random integrands

Facilitates numerical computation of integrals

Abstract

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two such extensions, corresponding to the \Ito-Skorokhod and the Stratononovich integrals, and provides an efficient tool to study these integrals, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields.