Multigraphs and Time Ordered Isserlis-Wick formulae

TL;DR

This paper introduces algebraic and diagrammatic methods, inspired by quantum field theory, to compute path-ordered exponentials of Gaussian processes with polynomial variables, enhancing analytical tools in stochastic analysis.

Contribution

It presents a novel diagrammatic approach using multigraph labelings for calculating path-ordered exponentials of Gaussian processes, complementing algebraic methods.

Findings

Developed an algebraic method for path-ordered exponentials.

Created a diagrammatic multigraph-based approach inspired by Feynman diagrams.

Provided explicit calculations demonstrating the methods' effectiveness.

Abstract

Given a m-dimensional Gaussian process and polynomial m variables with real coefficients, we calculate the induced path odered exponenial in two different ways: one is purely algebraic in spirit and the other one is diagrammatic in spirit and uses multigraph labelings (and is inspired by the use of Feynman diagrams in quantum field theory).