Orthogonal polynomials on path-space

TL;DR

This paper develops an orthogonal polynomial framework on path-space using the signature of stochastic processes, enabling function approximation on paths with convergence guarantees, and explores applications to Brownian motion.

Contribution

It introduces orthogonal polynomials on path-space via signature, proving density results and extending classical polynomial theory to stochastic processes.

Findings

Density of linear functions on signature in L^p spaces established

Orthogonal signature polynomials exist for Brownian motion with drift

Numerical examples demonstrate function approximation on paths

Abstract

We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in $L^p$ functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an $L^2$-convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.