Deterministic-like data-driven discovery of stochastic differential equations via the Feynman–Kac formalism

This paper develops a data-driven deterministic identification architecture for discovering stochastic differential equations (SDEs) directly from data. The architecture first generates deterministic data for stochastic processes using the Feynman–Kac formula, and gives a parabolic partial differential equation (PDE) associated with the SDE. Then, a sparse regression model is proposed to discover drift and diffusion terms in SDEs using PDE data-driven techniques, where a large candidate library of potential terms only for the drift and diffusion coefficients in SDEs need be constructed. To simultaneously infer the drift and diffusion terms, we proposed a sequential thresholded reweighted least-squares algorithm to solve the constructed sparse regression model. The main advantage of the proposed method is that on the one hand, theoretical and numerical identification results of PDEs can be used for SDEs, on the score, our SDE identification problem is translated into the parameter estimation problem of PDEs, on the other hand, the proposed algorithm is easily executed and can enhance the sparsity and accuracy. Through several classical SDEs and ordinary differential equations, the effectiveness of the proposed data-driven method is demonstrated, and several comparison experiments with state-of-the-art approaches is provided to illustrate the superiority of the developed algorithm.