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Modelling pathwise uncertainty of Stochastic Differential Equations samplers via Probabilistic Numerics

Yvann Le Fay, Simo Särkkä, Adrien Corenflos

TL;DR

This work extends probabilistic numerics to stochastic differential equations with additive noise by transforming Brownian motion into piecewise differentiable approximations and applying Gaussian ODE filters to obtain a pathwise probabilistic solution for $X_t$. It introduces two EKF-based SDE solvers and a marginalised variant, with rigorous convergence results for the EKF0-based schemes and a practical uncertainty calibration method for diffusion coefficients. The proposed methods yield strong convergence orders up to $1.5$ locally and $1.0$ globally (EKF0 on parabola Brownian approximations) and demonstrate robust weak convergence, validated on a stochastic oscillator model. The framework provides a principled way to quantify discretisation uncertainty in SDE simulations, at the cost of higher computation, and offers paths toward more accurate and calibrated stochastic solvers in applications requiring uncertainty-aware trajectory sampling.

Abstract

Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then queried on a grid to form a posterior distribution over the ODE solution. As the queries span the integration interval, the approximate posterior solution then converges to the true deterministic one. Gaussian ODE filters, in particular, have enjoyed a lot of attention due to their computational efficiency, the simplicity of their implementation, as well as their provable fast convergence rates. In this article, we extend the methodology to stochastic differential equations (SDEs) and propose a probabilistic simulator for SDEs. Our approach involves transforming the SDE into a sequence of random ODEs using piecewise differentiable approximations of the Brownian motion. We then apply probabilistic ODE solvers to the individual ODEs, resulting in a pathwise probabilistic solution to the SDE\@. We establish worst-case strong $1.5$ local and $1.0$ global convergence orders for a specific instance of our method. We further show how we can marginalise the Brownian approximations, by incorporating its coefficients as part of the prior ODE model, allowing for computing exact transition densities under our model. Finally, we numerically validate the theoretical findings, showcasing reasonable weak convergence properties in the marginalised version.

Modelling pathwise uncertainty of Stochastic Differential Equations samplers via Probabilistic Numerics

TL;DR

This work extends probabilistic numerics to stochastic differential equations with additive noise by transforming Brownian motion into piecewise differentiable approximations and applying Gaussian ODE filters to obtain a pathwise probabilistic solution for . It introduces two EKF-based SDE solvers and a marginalised variant, with rigorous convergence results for the EKF0-based schemes and a practical uncertainty calibration method for diffusion coefficients. The proposed methods yield strong convergence orders up to locally and globally (EKF0 on parabola Brownian approximations) and demonstrate robust weak convergence, validated on a stochastic oscillator model. The framework provides a principled way to quantify discretisation uncertainty in SDE simulations, at the cost of higher computation, and offers paths toward more accurate and calibrated stochastic solvers in applications requiring uncertainty-aware trajectory sampling.

Abstract

Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then queried on a grid to form a posterior distribution over the ODE solution. As the queries span the integration interval, the approximate posterior solution then converges to the true deterministic one. Gaussian ODE filters, in particular, have enjoyed a lot of attention due to their computational efficiency, the simplicity of their implementation, as well as their provable fast convergence rates. In this article, we extend the methodology to stochastic differential equations (SDEs) and propose a probabilistic simulator for SDEs. Our approach involves transforming the SDE into a sequence of random ODEs using piecewise differentiable approximations of the Brownian motion. We then apply probabilistic ODE solvers to the individual ODEs, resulting in a pathwise probabilistic solution to the SDE\@. We establish worst-case strong local and global convergence orders for a specific instance of our method. We further show how we can marginalise the Brownian approximations, by incorporating its coefficients as part of the prior ODE model, allowing for computing exact transition densities under our model. Finally, we numerically validate the theoretical findings, showcasing reasonable weak convergence properties in the marginalised version.
Paper Structure (21 sections, 7 theorems, 111 equations, 7 figures, 3 tables, 4 algorithms)

This paper contains 21 sections, 7 theorems, 111 equations, 7 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Outline and Contributions.
  3. Background
  4. Bayesian Uncertainty Quantification for ODEs
  5. The Gaussian ODE Filters
  6. Approximation of the Brownian Motion for Solving SDEs
  7. Modelling Uncertainty in SDEs
  8. The Gaussian SDE Filters
  9. A Gaussian Mixture Solution
  10. Marginalising the Brownian Approximation
  11. Uncertainty Calibration
  12. Experiments
  13. Experimental methodology
  14. Convergence Rates
  15. Uncertainty Calibration
  16. ...and 6 more sections

Key Result

Theorem 3.1

Assume the drift and diffusion functions satisfy Assumption assumption:reg_of_sde_flow, moreover, assume the Brownian approximation $\beta$ is given by the piecewise parabola eq:parabola. Let $X$ denote the solution of the SDE eq:additive_sde3. Let $\widetilde{X}$ denote the solution given by Algori where $||\cdot||_{\mathcal{L}^2}=\sqrt{\mathbb{E}[||\cdot||^2]}$ is the $\mathcal{L}^2$-norm and th

Figures (7)

  • Figure 1: A Brownian path and its linear and parabola approximations. The piecewise derivative of the parabola approximation is discontinuous at the $t_k$'s.
  • Figure 2: Algorithm \ref{['alg:ekf0-scheme2']}, Fitzhugh--Nagumo model. Independent IBM prior on each coordinates with $\eta=1$. Strong local and global errors, and weak global errors (grey points, black triangles and black squares, respectively) with log-log regression (grey, black line and dashed black line, respectively).
  • Figure 3: Algorithm \ref{['alg:ekf0-scheme3']}, Fitzhugh--Nagumo model. Strong, local and global errors, and weak global errors.
  • Figure 4: Fitzhugh--Nagumo model. Left: weak global errors for Algorithm \ref{['alg:ekf0-scheme2']} and \ref{['alg:ekf0-marginal-scheme']} using EKF0, in log-log scale. Right: trajectory means, from the fine solution (FS) and the estimated solution given by Algorithm \ref{['alg:ekf0-marginal-scheme']} using EKF1 with $\delta=2^{-10}$.
  • Figure 5: Fitzhugh--Nagumo model. Left: strong global errors for the Euler-Maruyama scheme with parameter $\delta$ and Algorithms \ref{['alg:ekf0-scheme2']} and \ref{['alg:ekf0-scheme3']}. Right: similar to the left but with weak global errors.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Remark 3.1
  • Theorem 3.1: Strong convergence of the Algorithm \ref{['alg:ekf0-scheme2']} using EKF0 linearisation
  • Remark 3.2
  • Theorem 3.2
  • Proposition 3.1: Quasi maximum-likelihood for the diffusion coefficient
  • Remark 3.3
  • Lemma 6.1
  • proof
  • Lemma 6.2: Kalman gain
  • proof
  • ...and 10 more