# MODELLING AND CALIBRATION OF STOCHASTIC PROCESSES WITH APPLICATION TO REAL DATA SETS

• Xiaodong Song

Student thesis: Phd

### Abstract

In this thesis, we build and calibrate models of stochastic processes with application to solar energy, finance and other fields. With population growth and technology development, the demand for electricity has increased dramatically. Due to the climate emergency including the greenhouse effect and depreciation of fossil fuels, renewable energy sources are encouraged by governmental policy and investment. Compared with other renewable sources, solar energy has the most potential around the world. Hence, accurate models are required that can provide not just solar power estimates but also capture the uncertainty in random processes. In Chapter 2, we propose a regime switching model of stochastic models (with jumps) for solar irradiance, and calibrate the model using solar data from Mauritius. Additionally, we develop a simulation method, which combines the Mycielski method with a Markov chain, to simulate and forecast future scenarios of solar irradiance. Based on historical data, our regime switching model and simulation method can give a good simulation and forecasting to the time variation of solar irradiance. We then derive the Fokker-Planck equation for a generalized Ornstein-Uhlenbeck process in Chapter 3. We present a Crank-Nicolson method to solve this (singular) version of the Fokker-Planck equation. Furthermore, we investigate the version proposed by Chang and Cooper, which has been used extensively in the past to solve numerical results for Fokker-Planck equations. We develop two improved Chang-Cooper methods, and compare these methods with our Crank-Nicolson method. We show that all three methods can give more accurate results and require less CPU time compared with Monte Carlo simulations, and our Crank-Nicolson method can give the most accurate results, but it requires more CPU time than two improved Chang-Cooper methods. In Chapter 4, we derive the Fokker-Planck equations with jumps to model the regime switching. We construct a partial integro-differential equation system, and then develop numerical schemes to solve this system. We then apply the numerical scheme to the solar irradiance data from Mauritius. We compare numerical results with Monte Carlo simulations, and we confirm the numerical results of the system can give a good estimation for the probability density function of the solar irradiance model, and it requires quite less CPU time than Monte Carlo simulations. In Chapter 5, we summarize the work and list the future work in jump size distributions, solar energy pricing and the 2-dimensional Fokker-Planck equation.
Date of Award 31 Dec 2021 English The University of Manchester Peter Duck (Supervisor) & Paul Johnson (Supervisor)

### Keywords

• Jumps
• Fokker-Planck equations
• Finite difference method
• Clustering
• Monte Carlo simulations
• Regime switching
• Stochastic differential equations
• Solar Power
• Scenario analysis

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