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Path-Dependent SDEs: Solutions and Parameter Estimation

Pardis Semnani, Vincent Guan, Elina Robeva, Darrick Lee

TL;DR

This work develops a general, history-aware framework for stochastic dynamics by introducing signature SDEs in which the drift and diffusion are linear functionals of the path signature $S(Y)$. Using rough path theory, the authors establish short-time existence and uniqueness for signature SDEs and formulate the lifted rough-path system that captures path dependence. They then propose the Expected Signature Matching Method (ESMM) to consistently estimate unknown parameters from trajectory data, proving convergence under differentiability and invertible Jacobian conditions and demonstrating exponential-rate behavior in the Picard-depth parameter. Empirical simulations across multiple dimensions show accurate parameter recovery and reveal the influence of the chosen signature terms on identifiability and estimation accuracy. The results push toward a general, signature-based parameter estimation framework applicable to both path-independent and path-dependent processes with rigorous theoretical guarantees and practical algorithms.

Abstract

We develop a consistent method for estimating the parameters of a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. We then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we prove that ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. Finally, we demonstrate on a variety of empirical simulations that our ESMM accurately infers the drift and diffusion parameters from observed trajectories. While parameter estimation is often restricted by the need for a suitable parametric model, this work makes progress toward a completely general framework for SDE parameter estimation, using signature terms to model arbitrary path-independent and path-dependent processes.

Path-Dependent SDEs: Solutions and Parameter Estimation

TL;DR

This work develops a general, history-aware framework for stochastic dynamics by introducing signature SDEs in which the drift and diffusion are linear functionals of the path signature . Using rough path theory, the authors establish short-time existence and uniqueness for signature SDEs and formulate the lifted rough-path system that captures path dependence. They then propose the Expected Signature Matching Method (ESMM) to consistently estimate unknown parameters from trajectory data, proving convergence under differentiability and invertible Jacobian conditions and demonstrating exponential-rate behavior in the Picard-depth parameter. Empirical simulations across multiple dimensions show accurate parameter recovery and reveal the influence of the chosen signature terms on identifiability and estimation accuracy. The results push toward a general, signature-based parameter estimation framework applicable to both path-independent and path-dependent processes with rigorous theoretical guarantees and practical algorithms.

Abstract

We develop a consistent method for estimating the parameters of a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. We then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we prove that ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. Finally, we demonstrate on a variety of empirical simulations that our ESMM accurately infers the drift and diffusion parameters from observed trajectories. While parameter estimation is often restricted by the need for a suitable parametric model, this work makes progress toward a completely general framework for SDE parameter estimation, using signature terms to model arbitrary path-independent and path-dependent processes.

Paper Structure

This paper contains 19 sections, 20 theorems, 170 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path Signatures in the Young Regime
  4. Rough Paths
  5. Universal Limit Theorem
  6. Universal and Characteristic Properties
  7. Path-Dependent Differential Equations
  8. Reformulation with Modified Vector Field
  9. The Solution as a Geometric Rough Path
  10. Parametrized Signature SDEs
  11. Expected Signature Matching Method
  12. Picard Iterations as Polynomials of Parameters
  13. Empirical Estimator
  14. Consistency
  15. Experiments
  16. ...and 4 more sections

Key Result

Theorem 1.1

(Informal) If $\mathbold{X}$ is an appropriately bounded rough path, then there exists a unique solution $\mathbold{Y}$ to eq:intro_signature_cde, and there exists a sequence of Picard iterations $\{\mathbold{Y}(r)\}_{r\in \mathbb{Z}_{\geq 0}}$, where $\mathbold{Y} = \lim_{r \to \infty} \mathbold{Y}

Figures (4)

  • Figure 1: The underlying paths in the sampled solutions to the path-dependent SDEs in \ref{['eq: non-identifiablility example 1']} and \ref{['eq: non-identifiablility example 2']} are plotted for each component of the paths over the interval $[0,T]$ for (a) $T=0.3$ and (b) $T=2.0$. The $L^2$ distance between the two trajectories evaluated at $T$ is (a) $0.045$ and (b) $1.743$.
  • Figure 2: The components of all the real solutions to the polynomial system \ref{['eq: expected signature mathcing method - modified']} over 100 trials in \ref{['experiment: 1 dimensional']} corresponding to the sets of words (a) $\mathcal{W}_1$ and (b) $\mathcal{W}_2$, defined in \ref{['eq: set of words 1']}.
  • Figure 3: The components of all the real solutions to the polynomial system \ref{['eq: expected signature mathcing method - modified']} over 100 trials in \ref{['experiment: 2 dimensional']} corresponding to the sets of words (a) $\mathcal{W}_1$ and (b) $\mathcal{W}_2$, defined in \ref{['eq: set of words 2']}.
  • Figure 4: The components of all the real solutions to the polynomial system \ref{['eq: expected signature mathcing method - modified']} over 100 trials in \ref{['experiment: 3 dimensional']} corresponding to the sets of words (a) $\mathcal{W}_5$ and (b) $\mathcal{W}_6$, defined in \ref{['eq: set of words 3']}. Only a few of the estimated values for $\theta^5$ fall within the interval $[-2,2]$, and thus, the rest are not visible in these plots. In each trial, many of the estimated values for $\theta^2$, $\theta^4$, and $\theta^6$ nearly coincide.

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Corollary 2.8
  • ...and 56 more