| This dissertation is divided into three parts.; In the first part, we consider two numerical schemes for deterministic differential equations: the Adams-Bashforth scheme and an “implicit leapfrog” scheme used to simulate a certain barotropic vorticity equation. We derive stochastic analogues of these schemes and prove their convergence to solutions of corresponding Itô stochastic differential equations.; In the second part, we consider the methods that Mil'shtein used to prove the convergence of the numerical scheme now named for him to the solution of an Itô stochastic differential equation. We then use his methods to prove the convergence of various schemes related to Mil'shtein's scheme to Itô, Stratanovich, and other general stochastic differential equations.; In the third part, we consider the so-called primitive equations for the atmosphere and prove maximum principles for the solution to these equations using both a classical method and the method of Stampacchia. |
