Dissertation Defense
Matching Problems for Stochastic Processes, by Joshua Beal (Ohio University, Mathematics)
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When 
Apr 23, 2013 from 03:30 PM to 05:30 PM 
Where  322 Morton Hall 
Contact Name  Archil Gulisashvili 
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Abstract: This dissertation investigates the nature of the 2D mimicking or matching problem, where the objective is to show the existence of a stochastic process Y that has the same joint (2D) distributions as a given stochastic process X. Typically, the goal is to ‘match’ X with a process Y having desirable properties such as the Markov and/or the martingale property. Much attention has already been given to 1D matching problems where the aim is to match X with a process Y having the same one dimensional (1D) distributions, or marginals, as X. The discussion below addresses the more general notion of matching in 2 or N dimensions. We generalize a theorem due to Strassen that provides conditions under which given probability measures are the marginals of some distribution lying in a closed convex set. Specifically, our generalization provides conditions under which given probability measures agree with the 2D distributions of a process of projections. Following this generalization, we apply our theorem to prove statements related to the existence of martingales with predefined 2D distributions. We demonstrate that the 2D matching problem is sufficiently robust by providing an example of a discrete time (nonmartingale) process matching the 2D distributions of a martingale. An ordering schema is also given which implies necessary conditions for 2D matching. In the 1D setting, we give conditions under which the product measure \(\mu \times \nu\) is a martingale measure, and later analyze problems related to local martingales. Lastly, we show that the space of ND distributions of an Nstep process X with values in a finite set is the convex hull of all measures for which X is Markov.