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Noise estimation of SDE from a single data trajectory

Munawar Ali, Purba Das, Qi Feng, Liyao Gao, Guang Lin

TL;DR

This work tackles the problem of learning stochastic differential equation dynamics from a single non-ergodic trajectory. It develops a data-driven pipeline that combines stochastic Taylor expansions with Girsanov transformations to estimate the drift $\mu(\cdot)$, recover the diffusion $\sigma(\cdot)$, and reconstruct the underlying Brownian increments $B^{\mathbb{P}}$, all from a single path. The Stochastic Sparse Identification of SDEs (SSISDE) algorithm then performs sparse symbolic regression to identify the governing SDE in terms of $\mu$ and $\sigma$, even when only one trajectory is available. The method is validated on linear and quadratic drift–diffusion models, including the non-ergodic Black-Scholes SDE, demonstrating accurate recovery of drift, diffusion, and noise with a parsimonious model. This approach enables practical data-driven stochastic model discovery in settings like finance where multiple trajectories are not available.

Abstract

In this paper, we propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By combining (stochastic) Taylor expansions with Girsanov transformations, and using the drift function's initial value as input, we construct drift estimators while simultaneously recovering the model noise. This allows us to recover the underlying $\mathbb P$ Brownian motion increments. Building on these estimators, we introduce the first stochastic Sparse Identification of Stochastic Differential Equation (SSISDE) algorithm, capable of identifying the governing SDE dynamics from a single observed trajectory without requiring ergodicity or stationarity. To validate the proposed approach, we conduct numerical experiments with both linear and quadratic drift-diffusion functions. Among these, the Black-Scholes SDE is included as a representative case of a system that does not satisfy ergodicity or stationarity.

Noise estimation of SDE from a single data trajectory

TL;DR

This work tackles the problem of learning stochastic differential equation dynamics from a single non-ergodic trajectory. It develops a data-driven pipeline that combines stochastic Taylor expansions with Girsanov transformations to estimate the drift , recover the diffusion , and reconstruct the underlying Brownian increments , all from a single path. The Stochastic Sparse Identification of SDEs (SSISDE) algorithm then performs sparse symbolic regression to identify the governing SDE in terms of and , even when only one trajectory is available. The method is validated on linear and quadratic drift–diffusion models, including the non-ergodic Black-Scholes SDE, demonstrating accurate recovery of drift, diffusion, and noise with a parsimonious model. This approach enables practical data-driven stochastic model discovery in settings like finance where multiple trajectories are not available.

Abstract

In this paper, we propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By combining (stochastic) Taylor expansions with Girsanov transformations, and using the drift function's initial value as input, we construct drift estimators while simultaneously recovering the model noise. This allows us to recover the underlying Brownian motion increments. Building on these estimators, we introduce the first stochastic Sparse Identification of Stochastic Differential Equation (SSISDE) algorithm, capable of identifying the governing SDE dynamics from a single observed trajectory without requiring ergodicity or stationarity. To validate the proposed approach, we conduct numerical experiments with both linear and quadratic drift-diffusion functions. Among these, the Black-Scholes SDE is included as a representative case of a system that does not satisfy ergodicity or stationarity.

Paper Structure

This paper contains 15 sections, 60 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and problem formulation
  3. Main Results
  4. Proof of Step 2.
  5. Proof of Step 3.
  6. Proof of Step 4.
  7. Proof of Step 5
  8. Numerical Experiments
  9. Model description
  10. Model selection via TEXT-fold time–series cross-validation
  11. Choice of the final model
  12. Optimally sparse size
  13. Practical settings used in our experiments
  14. Black-Scholes model
  15. Quadratic coefficients SDE

Figures (6)

  • Figure 1: Time–series CV over $(\alpha,\rho)$ with de-biasing on the active set; the selected point $(\alpha^\dagger,\rho^\dagger)=(3.87\times 10^{-4},\,0.85)$ lies on a broad near-optimal basin.
  • Figure 2: Validation error $\delta(n)$ vs. support size $n$ with $\pm1$ SE bands for the drift and diffusion blocks. The vertical markers indicate the optimally sparse sizes and the sizes of the final full-data models.
  • Figure 3: (a) The true $\mu(X_t)$ and reconstructed $\widehat{\mu}(X_t)$ and (b) The true $\sigma(X_t)$ and reconstructed $\widehat{\sigma}(X_t)$ are plotted against time on the interval $[0,1]$. (c) The true noise, i.e., Brownian motion $B^{\mathbb{P}}$ that was used to simulate the actual path, the recovered noise, and the $\mathbb{Q}$-Brownian motion are plotted against time. (d) The actual path of the data $X_t$ and the reconstructed one from the $\widehat{\mu}(X_t)$ and $\widehat{\sigma}(X_t)$ are plotted against time. Each vector is of length $M$ with $M=1000$.
  • Figure 4: Time–series cross-validation over $(\alpha,\rho)$ with de-biasing on the active set. The selected point $(\alpha^\dagger,\rho^\dagger)=(8.73\times 10^{-4},\,0.50)$ lies on a broad plateau of optimal values.
  • Figure 5: Validation error $\delta(n)$ vs. support size $n$ with $\pm 1$ SE bands. Vertical lines indicate the optimally sparse sizes $\tilde{n}$ (1-SE rule) and the final full-data sizes.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Remark 3.1