Variational inference of the drift function for stochastic differential equations driven by L\'{e}vy processes

TL;DR

This paper introduces a variational approach to nonparametrically estimate the drift function of stochastic differential equations driven by Lévy processes, utilizing path probability divergence and empirical data.

Contribution

It develops a novel variational formula based on the stationary Fokker-Planck equation for Lévy-driven SDEs, enabling nonparametric drift estimation from data.

Findings

Estimation accuracy improves with more data.

Higher α values lead to better drift estimation.

The method effectively captures the drift function in simulations.

Abstract

In this paper, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by $\alpha$-stable L\'{e}vy motion. First, the Kullback-Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions is optimized. By using the Lagrangian multiplier, the variational formula based on the stationary Fokker-Planck equation is constructed. Then combined with the data information, the empirical distribution is used to replace the stationary density, and the drift function is estimated non-parametrically from the perspective of the process. In the numerical experiment, the different amounts of data and different $\alpha$ values are studied. The experimental results show that the estimation result of the drift function is related to both. When the amount of data increases, the estimation result will be better, and when the $\alpha$ value increases, the estimation result is also better.