Noise estimation of SDE from a single data trajectory

ArXiv ID: 2509.25484 “View on arXiv”

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

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.

Keywords: stochastic differential equations (SDEs), stochastic sparse identification, Girsanov transformation, Taylor expansion, single trajectory estimation, General Equities

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, utilizing advanced stochastic calculus (Itô/Stratonovich calculus, Girsanov transformation, stochastic Taylor expansions, rough path theory) and requires a strong theoretical foundation. While it includes numerical experiments (e.g., Black-Scholes SDE) to validate the algorithm, it focuses on model discovery and estimation from a single trajectory rather than portfolio backtesting, transaction costs, or live trading data.

  flowchart TD
    A["Research Goal<br>Estimate SDE noise & drift<br>from a single trajectory<br>without ergodicity"] --> B{"Data Input"}
    B --> C["Single Observed Trajectory<br>e.g., Black-Scholes Market Data"]
    B --> D["Drift Function Initial Value"]

    C & D --> E["Methodology<br>Stochastic Taylor Expansion<br>+ Girsanov Transformation"]
    
    E --> F["Computational Process<br>Stochastic Sparse Identification<br>of SDE SSISDE"]
    
    F --> G["Key Outcomes"]
    
    subgraph G [" "]
        G1["Recovered Drift Function"]
        G2["Estimated Model Noise<br>Recovering P Brownian Motion"]
        G3["Validated on Non-Ergodic Systems<br>e.g., Black-Scholes"]
    end