Access Type

Open Access Dissertation

Date of Award

January 2016

Degree Type


Degree Name




First Advisor

George Yin


This dissertation contains two main parts. The first part focuses on numerical algorithms for approximating the ergodic means of suitable functions of solutions to stochastic differential equations with Markov regime switching. Our main effort is devoted to obtaining the convergence and rates of convergence of the approximation algorithms. The study is carried out by obtaining laws of large numbers and laws of iterated logarithms for numerical approximation to long-run averages of suitable functions of solutions to switching diffusions.

The second part is devoted to stochastic functional differential equations (SFDEs) with infinite delay. This part consists of two main themes. First, existence and uniqueness of the solutions of such equations are examined. Because the solutions of the delay equations are not Markov, a viable alternative for studying further asymptotic properties

is to use solution maps or segment processes. By examining solution maps, we investigates the Markov properties as well as the strong Markov properties. Also obtained are adaptivity and continuity, mean-square boundedness, and convergence of solution maps from differential initial data. Then we examine the ergodicity of underlying processes and establishes existence of the invariant measure for SFDEs with infinite delay under suitable conditions.

The next theme examines the properties of stochastic integro-differential equations with infinite delay (or unbounded delay). Our main approach is to map the solution processes into another Polish space. Under suitable conditions, it is shown that the resulting processes are Markov. Furthermore, sufficient conditions for Feller property, recurrence, ergodicity, and existence of invariant measures are obtained. Moreover, weak sense Fokker-Planck equations are derived for the underlying processes.

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