Variational inference of the drift function for stochastic differential equations driven by Lévy processes

In this work, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by the α-stable Lévy process. We first optimize the Kullback-Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions. We then construct the variational formula based on the stationary Fokker-Planck equation using the Lagrangian multiplier. Moreover, we apply the empirical distribution to replace the stationary density, combining it with the data information, and we present the estimator of the drift function from the perspective of the process. In the numerical experiment, we investigate the effect of the different amounts of data and different α values. The experimental results demonstrate that the estimation result of the drift function is related to both and that the exact drift function agrees well with the estimated result. The estimation result will be better when the amount of data increases, and the estimation result is also better when the α value increases.