Multidimensional Jump-diffusion Processes

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Hamounian, Abolfazl | 2023

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 56864 (04)
University: Sharif University of Technology
Department: Physics
Advisor(s): Rahimi Tabar, Mohammad Reza
Abstract:
In this thesis, we first provide a brief overview of the basic concepts of stochastic processes and the mathematical tools required to study these processes. Then, we demonstrate that by using a data-driven approach, one can obtain the corresponding coefficients for deterministic, diffusion, and jump components of the stochastic process. In this method, the conditional moments of Kramers–Moyal coefficients are calculated using statistical moments of N-dimensional time series. These coefficients enable us to determine different orders of interactions, diffusion and jump properties, and %relationships coupling coefficients between time series. Furthermore, to evaluate the accuracy of the proposed method, we examine several dynamical equations in different dimensions, and finally, we analyze a few examples of real two-dimensional time series.
Keywords:
Stochastic Process ; Langevin Equation ; Jump-Diffusion Model ; Kramers-Moyal Coefficient ; Time Series Analysis