Thesis (Ph. D.)--University of Rochester. Dept. of Mathematics, 2013.
The theory of stochastic differential equations (SDE) describes the world using differential
equations, including randomness as a fundamental factor. This theory integrates
randomness into the equations using Ito's theory of stochastic calculus allowing
to study the usual wave or heat equation, accounting for unknown events that can
modify the solutions.
This work contains three major parts. The fi rst part proves uniqueness of the
solution to a stochastic differential equation. This equation has relations with the
wave equation and is a first attempt to prove uniqueness for a stochastic partial
differential equation. The second part focuses on a new method to prove uniqueness
for stochastic partial differential equations. This method transforms the question
of uniqueness from the stochastic partial differential equation to a doubly backward
stochastic differential equation. The third part is the study of an equivalence relation
of binary matrices. I develop an algorithm to find the representative of each class of
binary matrices.
