Matching Problems for Stochastic Processes, by Joshua Beal (Ohio University, Mathematics)
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 2-D mimicking or matching problem, where the objective is to show the existence of a stochastic process Y that has the same joint (2-D) 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 1-D matching problems where the aim is to match X with a process Y having the same one dimensional (1-D) 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 2-D distributions of a process of projections. Following this generalization, we apply our theorem to prove statements related to the existence of martingales with pre-defined 2-D distributions. We demonstrate that the 2-D matching problem is sufficiently robust by providing an example of a discrete time (non-martingale) process matching the 2-D distributions of a martingale. An ordering schema is also given which implies necessary conditions for 2-D matching. In the 1-D 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 N-D distributions of an N-step process X with values in a finite set is the convex hull of all measures for which X is Markov.