Orthogonal polynomials on path-space

2026-02-21

AI Summary

This work addresses the construction of orthogonal bases on path space analogous to classical orthogonal polynomials, enabling efficient representation and approximation of square-integrable functionals of stochastic paths. By orthogonalizing the signature of stochastic processes within the shuffle algebra and free Lie algebra frameworks and analyzing $L^2$ convergence, the study extends multivariate orthogonal polynomial theory—including recurrence relations and Favard’s theorem—to path space for the first time. Key contributions include demonstrating the existence of dimension-free orthogonal signatures for Brownian motion with drift, in contrast to the driftless case where such signatures do not exist; establishing the $L^p$-density of linear signature functionals over group-like elements; and constructing orthogonal signature polynomials for Brownian motion, whose efficacy in approximating path functionals under Wiener measure is validated through numerical experiments.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Other Foundations of Search & Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• orthogonal polynomials
• path-space
• signature
• Brownian motion
• shuffle algebra

Innovation

• orthogonal signature
• shuffle algebra
• path-space orthogonal polynomials
• Brownian motion with drift
• Favard's theorem