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We investigate parameter estimation for stochastic differential equations driven by fractional Lévy processes. Our study focuses on the consistency and asymptotic behavior of minimum distance and minimum norm estimators.

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Area of Science:

  • Stochastic Analysis
  • Probability Theory
  • Time Series Analysis

Background:

  • Stochastic differential equations (SDEs) are fundamental in modeling complex systems.
  • Fractional Lévy processes introduce long-range dependence and jumps, enhancing SDE modeling capabilities.
  • Parameter estimation is crucial for understanding and predicting the behavior of SDEs.

Purpose of the Study:

  • To analyze the minimum Skorohod distance estimation and minimum L2-norm estimation for the drift parameter of SDEs.
  • To investigate the statistical properties of these estimators for a specific class of SDEs.
  • To examine the asymptotic behavior of the estimators under different conditions.

Main Methods:

  • Utilizing Skorohod distance and L2-norm for parameter estimation.
  • Deriving consistency and limit distributions of the estimators.
  • Analyzing asymptotic laws for large time horizons.

Main Results:

  • Established the consistency of both minimum Skorohod distance and minimum L2-norm estimators.
  • Determined the limit distributions for fixed time T as the noise intensity approaches zero.
  • Investigated the asymptotic laws governing these distributions for large time scales.

Conclusions:

  • The proposed estimation methods are statistically sound for SDEs driven by fractional Lévy processes.
  • The findings provide theoretical guarantees for parameter estimation in these complex models.
  • This research contributes to the robust analysis of stochastic systems with non-standard noise characteristics.