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Matching Problems for Stochastic Processes

Beal, Joshua M
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367500889

Abstract Details

2013, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
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 suciently 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 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.
Archil Gulisashvili, Dr. (Advisor)
79 p.
Mathematics
martingale; Markov; stochastic process

Recommended Citations

Citations

  • Beal, J. M. (2013). Matching Problems for Stochastic Processes [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367500889

    APA Style (7th edition)

  • Beal, Joshua. Matching Problems for Stochastic Processes. 2013. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367500889.

    MLA Style (8th edition)

  • Beal, Joshua. "Matching Problems for Stochastic Processes." Doctoral dissertation, Ohio University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367500889

    Chicago Manual of Style (17th edition)

ohiou1367500889
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This open access ETD is published by Ohio University and OhioLINK.