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

Area of Science:

• Stochastic Analysis
• Nonparametric Statistics
• Lévy Processes

Background:

• Stochastic differential equations (SDEs) are crucial for modeling complex systems.
• Estimating drift functions in SDEs driven by Lévy processes presents unique challenges.
• Alpha-stable Lévy processes offer a more flexible framework than Brownian motion for modeling phenomena with heavy tails.

Purpose of the Study:

• To develop a nonparametric method for estimating the drift function of SDEs driven by alpha-stable Lévy processes.
• To optimize the Kullback-Leibler divergence for improved estimation accuracy.
• To provide a robust estimation technique applicable to real-world data.

Main Methods:

• Optimization of Kullback-Leibler divergence between path probabilities of SDEs.
• Construction of a variational formula using the stationary Fokker-Planck equation and a Lagrangian multiplier.
• Application of empirical distribution to approximate stationary density, integrating data information.
• Estimation from the perspective of the stochastic process itself.

Main Results:

• The proposed method effectively estimates the drift function of SDEs driven by alpha-stable Lévy processes.
• Estimation accuracy improves with an increase in the amount of data.
• Higher values of the parameter alpha lead to better estimation results.
• The estimated drift functions show good agreement with the exact drift functions in numerical experiments.

Conclusions:

• The developed nonparametric method provides a reliable approach for drift function estimation in alpha-stable SDEs.
• Data quantity and the alpha parameter significantly influence the estimation performance.
• This work contributes to the theoretical understanding and practical application of SDEs with heavy-tailed noise.