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Dimension reduction for path signatures

Christian Bayer, Martin Redmann

TL;DR

The paper addresses the high-dimensional complexity of path-signature models by developing a projection-based model order reduction framework tailored to signatures of continuous semimartingales. It rewrites signature dynamics as an Itô system, constructs Gram-type matrices $P$ and $Q$ to identify dominant and negligible directions, and uses a balancing transformation to obtain a reduced-order model that preserves the linear-output map of interest. An algebraic error representation enables a priori assessment of MOR accuracy without Monte Carlo evaluation. Numerical experiments on Bergomi and rough Bergomi models demonstrate substantial dimensionality reduction (e.g., from $n=1365$ to $\tilde{n}=27$ or $\tilde{n}=55$) with high fidelity in option-implied volatilities, underscoring the practical viability for complex, memory-bearing financial models.

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.

Dimension reduction for path signatures

TL;DR

The paper addresses the high-dimensional complexity of path-signature models by developing a projection-based model order reduction framework tailored to signatures of continuous semimartingales. It rewrites signature dynamics as an Itô system, constructs Gram-type matrices and to identify dominant and negligible directions, and uses a balancing transformation to obtain a reduced-order model that preserves the linear-output map of interest. An algebraic error representation enables a priori assessment of MOR accuracy without Monte Carlo evaluation. Numerical experiments on Bergomi and rough Bergomi models demonstrate substantial dimensionality reduction (e.g., from to or ) with high fidelity in option-implied volatilities, underscoring the practical viability for complex, memory-bearing financial models.

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.

Paper Structure

This paper contains 17 sections, 8 theorems, 75 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Signatures
  3. Dimension reduction
  4. Identifying less relevant signature information
  5. Computation of $P$ and $Q$
  6. Reduced order signature approximation
  7. Algebraic error representation
  8. Numerical examples
  9. Bergomi model
  10. Rough Bergomi model
  11. Introduction to signatures
  12. Signatures of smooth paths
  13. Signatures for Hölder continuous paths
  14. Signatures of semimartingales
  15. Universal approximation theorem
  16. ...and 2 more sections

Key Result

Proposition 3.4

Let $(\lambda_k)_{k=1, \dots, n}$ be the eigenvalues corresponding to the ONB $(p_k)_{k=1, \dots, n}$. Then, we have that

Figures (8)

  • Figure 1: Square root of first $50$ out of $n=1365$ eigenvalues of $PQ$ for signature model associated with \ref{['bergomi_model']}.
  • Figure 2: $L^2$-error between output of the signature model of \ref{['bergomi_model']} ($n=1365$) and reduced system output for $\widetilde{n}=1, \dots, 26$.
  • Figure 3: Implied volatilities of the signature-based approximation $S^{(\ell)}$ of \ref{['bergomi_model']} for $T=1/12, 1/2, 1$ and strike prices $K_T=(0.8+j\cdot 0.02)^{\sqrt{T}}$ with $j=0, 1, \dots, 20$.
  • Figure 4: Relative error between volatilities of signature model associated with \ref{['bergomi_model']} and reduced systems with $\widetilde{n}=5, 11, 27$, strike prices $K_T=(0.8+j\cdot 0.02)^{\sqrt{T}}$ and $T=1/12, 1/2, 1$.
  • Figure 5: Square root of first $80$ out of $n=3280$ eigenvalues of $PQ$ for signature model associated with \ref{['rough_bergomi_model']}.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Remark 3.7
  • ...and 18 more