Dimension reduction for path signatures

TL;DR

This paper develops methods to reduce the complexity of path signature models, which are used in stochastic differential equations and financial modeling, by lowering their order while preserving key features.

Contribution

It introduces novel techniques for reducing the order of high-dimensional signature models, maintaining their essential properties and applicability.

Findings

Reduced models effectively approximate original signatures

Numerical examples demonstrate the methods' effectiveness

Applications include the rough Bergomi model in finance

Abstract

This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and their applications in areas such as stochastic differential equations (SDEs) and financial modeling are pointed out. In particular, exploiting the rough paths view, solutions of SDEs continuously depend on the lift of the driver. Such continuous mappings can be approximated using (truncated) signatures, which are solutions of high-dimensional linear systems. In order to lower the complexity of these models, this paper presents methods for reducing the order of high-dimensional truncated signature models while retaining essential characteristics. The derivation of reduced models and the universal approximation property of (truncated) signatures are treated in detail. Numerical examples, including applications to the (rough) Bergomi model in financial markets, illustrate the proposed reduction techniques and highlight their effectiveness.