Xu, Shijie 
ORCID: 0000-0001-8293-4509
(2024)
Modelling the term structure of energy markets and some density bounds for Itô processes.
PhD thesis, University of Liverpool.
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Abstract
This thesis is concerned with the modelling of the term structure of energy forward contracts and upper estimates for the density or occupation of a diffusion. First, we study finite-dimensional models for the entire term structure of energy forwards. Once a finite-dimensional set of possible yield curves is chosen, we estimate the dynamic behaviour of the yield curve evolution using data. The estimated model should be free of arbitrage, which is known to result in some drift condition. The diffusion coefficient is then estimated from data. From a practical perspective, this requires that the obtained diffusion coefficient be compatible with the set of a priori chosen yield curves. This implies that we must have at least an open set of possible diffusion coefficients that are compatible with our model. We show that such compatibility, which we call the statistical consistency condition (SCC), enforces an affine geometry on the set of possible yield curves. We identify all functions g such that if we first perform static fitting and then dynamic fitting, it automatically leads to an arbitrage-free model. By static fitting, we mean fitting the model to a fixed dataset. By dynamic fitting, we mean identifying this stochastic process over time. In the first project, we encountered the question of whether we have density for the marginals of a diffusion process while we only had local information on the drift and constant diffusion coefficient. This led to the question of whether we can indeed find explicit upper bounds for the marginal density. In our second project, we derive explicit upper bounds for the density of marginals of continuous diffusions where we assume that the diffusion coefficient is constant and the drift is solely assumed to be progressively measurable and locally bounded. In 1-dimension we extend our result to the case that the diffusion coefficient is a locally Lipschitz- continuous function of the state. Our approach is based on a comparison to a suitable doubly reflected Brownian motion, whose density is known in a series representation. The second project left the obvious question about what happens if the diffusion coefficient is non-constant. It is known that in this case, marginal densities do not need to exist. In the third project, we find explicit and optimal upper "time-average" bounds for the densities. Unlike the density of the marginals, its "time-average" always exists in a generalised sense, namely the Lebesgue density of the expected occupation measure. Our findings are related to the optimal bound for the expected interval occupation found in Ankirchner and Wendt (2021). In contrast, our bound is for a single point, and the resulting formula is less involved. Our findings allow us to find explicit upper bounds for mean path integrals.
| Item Type: | Thesis (PhD) |
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| Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 01 Oct 2024 11:19 |
| Last Modified: | 08 Feb 2025 03:06 |
| DOI: | 10.17638/03184228 |
| Supervisors: |
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| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3184228 |
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