Data-driven Feynman-Kac Discovery with Applications to Prediction and Data Generation
Qi Feng, Guang Lin, Purav Matlia, Denny Serdarevic
TL;DR
The paper addresses learning probabilistic laws underlying the Feynman–Kac representation by recovering the backward SDE associated with option pricing from a single stock–option data pair under the risk-neutral measure. It introduces a stochastic SINDy framework that jointly learns a neural approximation of $u(t,x)$, estimates the diffusion, extracts Brownian increments under $\mathbb Q$, and identifies sparse drivers $f$ and $Z$ in the BSDE $Y_t=g(X_T)+\int_t^T f(\cdot)\,ds - \int_t^T Z_s\,dB^{\mathbb Q}_s$. The method enables forward prediction of $Y_t$ and generation of synthetic trajectories that conform to the learned probabilistic law, demonstrated with Black–Scholes and real Apple data. By removing ergodicity requirements and applying stochastic SINDy to BSDEs, the approach opens a path toward data-driven discovery of Feynman–Kac–type laws in physics and engineering from limited time-series observations.
Abstract
In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law.