Dimension reduction for path signatures
Authors
- Bayer, Christian
ORCID: 0000-0002-9116-0039 - Redmann, Martin
ORCID: 0000-0001-5182-9773
2020 Mathematics Subject Classification
- 60H10 60L10 60L90
Keywords
- Signature models, applications of rough analysis, stochastic differential equations, model order reduction, Petrov-Galerkin projections, financial models
DOI
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 approxi- mation 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.
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