In the first part of the thesis, we focus on the development of a generalized It\^o stochastic differential equation (SDE) model to assess the uncertainty of deterministic forecasts when data are naturally constrained. By incorporating a time-dependent upper bound, we extend the It\^o Stochastic Differential Equation (SDE) framework to enable the path-wise evaluation of short-term forecast errors. This upper bound restricts the range of normalized observable historical data and forecasts, enhancing the accuracy of the evaluation process. In our study, we present a novel generalized SDE model that is nonlinear and time-inhomogeneous. To enhance the calibration accuracy, this model incorporates a diffusion term with two additional parameters that surpass the limitations of previous approaches, resulting in an improved quality of calibration, and is driven by both the forecast and an upper bound constraint. We use an iterative multi-stage optimization procedure to approximate the likelihood function and calibrate the SDE model parameters using stochastic optimization. We maximize the likelihood using adaptive momentum and learning rates and improve the time of convergence by a factor of 90 to 100. We also introduce an adaptive time-stepping method to ensure that the numerical approximation of the process does not violate the natural bounds imposed by the physical and mathematical constraints of the modeled problem.
In the second part of the thesis, we focus on SDE modeling of electricity spot prices by implementing a mean-reverting jump-diffusion SDE with a constant diffusion term and a Poisson jump part. We develop a two-step iterative method to identify jumps present in the data and use maximum likelihood estimation to calibrate the model parameters. We also present a dataset of electricity spot prices and their forecast for Germany spanning from 2016 to 2018 to identify areas where the model may require further refinement to capture the underlying characteristics of electricity spot prices more comprehensively.
| Date of Award | Jun 9 2023 |
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| Original language | English (US) |
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| Awarding Institution | - Computer, Electrical and Mathematical Sciences and Engineering
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| Supervisor | Raul Tempone (Supervisor) |
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