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

Min Dai1, Jinqiao Duan2, Jianyu Hu3

  • 1School of Science, Wuhan University of Technology, Wuhan 430070, China.

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Summary
This summary is machine-generated.

This study estimates the drift function for stochastic differential equations driven by alpha-stable Lévy processes. Improved estimation accuracy is observed with increased data and higher alpha values.

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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.