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Analysis and data-driven reconstruction of bivariate jump-diffusion processes.

Leonardo Rydin Gorjão1,2,3,4, Jan Heysel1,2, Klaus Lehnertz1,2,5

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This study introduces a new bivariate jump-diffusion model and a data-driven method to estimate its parameters using Kramers-Moyal coefficients. This approach helps reconstruct complex processes from observed data.

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Area of Science:

  • * Stochastic processes
  • * Data-driven modeling
  • * Time series analysis

Background:

  • * Jump-diffusion processes are essential for modeling financial markets and other complex systems.
  • * Accurately estimating parameters of these processes is crucial for reliable predictions.
  • * Existing methods may struggle with the complexity of coupled bivariate systems.

Purpose of the Study:

  • * To introduce a novel bivariate jump-diffusion process.
  • * To develop a data-driven, nonparametric method for estimating Kramers-Moyal coefficients.
  • * To enable the reconstruction of underlying process aspects and parameter recovery.

Main Methods:

  • * Development of a bivariate jump-diffusion process with coupled diffusion and jump components.
  • * Implementation of a nonparametric estimation procedure for higher-order Kramers-Moyal coefficients (up to order 8).
  • * Validation using synthetic bivariate time series data from continuous and discontinuous processes.

Main Results:

  • * Successful reconstruction of key aspects of underlying jump-diffusion processes.
  • * Recovery of process parameters using the developed estimation procedure.
  • * Demonstrated the utility of Kramers-Moyal coefficients for parameter estimation.

Conclusions:

  • * The proposed data-driven method effectively estimates parameters for bivariate jump-diffusion models.
  • * The method aids in understanding complex stochastic systems.
  • * Limitations related to data scarcity and parameter disproportion were identified.