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Signature SDEs from an affine and polynomial perspective

Christa Cuchiero, Sara Svaluto-Ferro, Josef Teichmann

TL;DR

This work constructs a universal, path-dependent diffusion framework by lifting Itô SDEs to their signature processes on the extended tensor algebra. By embedding these lifts into affine and polynomial processes, the authors derive explicit Fourier-Laplace transform and moment formulas via tensor-algebra Riccati and linear ODEs, yielding convergent power-series representations computable in infinite or finite dimensions. The paper develops a rigorous duality-based foundation for affine and polynomial processes on group-like signature state spaces, and demonstrates the approach on both multi-dimensional and one-dimensional, real-analytic settings, including concrete Brownian and Jacobi examples. Numerical illustrations showcase practical schemes to compute the full law on path space and the expected signature, with potential implications for neural SDEs and signature-based learning models. Overall, the results provide a unifying, computable framework for analyzing path-dependent SDEs through their signature lifts, enabling explicit characterization of path-space laws and moments.

Abstract

Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on the duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process' marginals. The coefficients of these power series are solutions of extended tensor algebra valued Riccati and linear ordinary differential equations (ODEs), respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thus construct a class of stochastic processes which is universal (in a sense specified in the introduction) within Ito-diffusions with path-dependent characteristics and allows for an explicit characterization of the Fourier-Laplace transform and hence the full law on path space. The practical applicability of this affine and polynomial approach is illustrated by several numerical examples.

Signature SDEs from an affine and polynomial perspective

TL;DR

This work constructs a universal, path-dependent diffusion framework by lifting Itô SDEs to their signature processes on the extended tensor algebra. By embedding these lifts into affine and polynomial processes, the authors derive explicit Fourier-Laplace transform and moment formulas via tensor-algebra Riccati and linear ODEs, yielding convergent power-series representations computable in infinite or finite dimensions. The paper develops a rigorous duality-based foundation for affine and polynomial processes on group-like signature state spaces, and demonstrates the approach on both multi-dimensional and one-dimensional, real-analytic settings, including concrete Brownian and Jacobi examples. Numerical illustrations showcase practical schemes to compute the full law on path space and the expected signature, with potential implications for neural SDEs and signature-based learning models. Overall, the results provide a unifying, computable framework for analyzing path-dependent SDEs through their signature lifts, enabling explicit characterization of path-space laws and moments.

Abstract

Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on the duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process' marginals. The coefficients of these power series are solutions of extended tensor algebra valued Riccati and linear ordinary differential equations (ODEs), respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thus construct a class of stochastic processes which is universal (in a sense specified in the introduction) within Ito-diffusions with path-dependent characteristics and allows for an explicit characterization of the Fourier-Laplace transform and hence the full law on path space. The practical applicability of this affine and polynomial approach is illustrated by several numerical examples.
Paper Structure (35 sections, 27 theorems, 208 equations, 4 figures)

This paper contains 35 sections, 27 theorems, 208 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Affine and polynomial processes from a duality point of view
  4. Affine case
  5. Polynomial case
  6. Signature of continuous semimartingales
  7. Affine and polynomial processes on $T((\mathbb{R}^d))$
  8. State space and spaces of dual elements
  9. Martingale problems
  10. The affine case
  11. The polynomial case
  12. Signature SDEs and their signature process
  13. State space of signature processes, shuffle-compatible spaces of dual elements and entire functions
  14. Signature SDEs as affine and polynomial processes
  15. The affine case
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $X$ be given by eq:SigSDEintro. Then, its signature process $\mathbb{X}$ is an affine and polynomial process on the state space of group-like elements, which satisfies under several technical conditions, where ${\bm \psi}$ and ${\bf c}$ are solutions of the tensor-algebra valued Riccati ODE and linear ODE with $R$ and $L$ specified in eq:R and eq:L, respectively.

Figures (4)

  • Figure 1: ${\mathbb E}[\exp(-\exp(X_t))]$ computed with Scheme \ref{['sch1']} for $K=20$.
  • Figure 2: ${\mathbb E}[\exp(-X_t^4/4!)]$ computed with Scheme \ref{['sch1']} for different $K$s. The approximations ${\mathbb E}[\exp(-X_t^4/4!)]$ of are just represented until their explosion time.
  • Figure 3: ${\mathbb E}[\exp(-X_t^4/4!)]$ computed with Scheme \ref{['sch2']} for $N=80$, $K=160$, and different values of $M$s. The approximations are just represented until their explosion time.
  • Figure 4: ${\mathbb E}[\exp(cX_T)]$ for a Jacobi diffusion with $X_0=\frac{1}{2}$ and $T=1000$ computed with Scheme \ref{['sch3']} for different truncation levels $K$. The result is compared with the moment generating function of the stationary measure $(\delta_0+\delta_1)/2$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 73 more