Kassis, Georges;
(2025)
Strong stochastic interpolation and informed martingale transport.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
Arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between zeros at fixed pre-specified times. Their additive randomisation allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomised arcade processes (RAPs) can thus be interpreted as a generalisation of anticipative stochastic bridges. The filtrations generated by these processes are utilised to construct a class of martingales which interpolate between the given target random variables. These so-called filtered arcade martingales (FAMs) are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally-Markov randomised arcade processes, the dynamics of FAMs are informed by Bayesian updating. To derive their dynamics, we had to determine the expression of the quadratic variation of a Gauss-Markov semimartingale. As an application of this theory in the field of optimal transportation, FAMs may be used to introduce noise in martingale optimal transport, in a similar fashion to how Schrödinger's problem introduces noise in optimal transport. This information-based approach to transport is concerned with selecting an optimal martingale coupling for the target random variables under the influence of the noise that is generated by an arcade process. This optimisation problem, that we called the information-based martingale optimal transport problem (IB-MOT), is static in its nature, since it is concerned with finding a coupling, but a corresponding dynamical solution can be found by considering the FAM constructed with the optimal coupling. Existence and uniqueness of its solution are shown, and a gradient-based algorithm for empirical measures is discussed.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Strong stochastic interpolation and informed martingale transport |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10211848 |
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