An investigation of numerical methods preserving boundedness of the solutions to the stochastic differential equations

Stochastic Differential Equations are treated in a different category from Deterministic Differential Equations because of the noise term. When coefficients of Stochastic Differential Equations satisfy global Lipschitz condition and linear growth condition, the exact solution to the Stochastic Differential Equation and the numerical solution obtained by explicit Euler- Maruyama scheme have bounded moments. If the conditions are relaxed, it is proved by Hutzenthaler et. al. in 2011 that the numerical solution obtained by explicit Euler- Maruyama scheme explodes although the exact solution stays bounded. In the following years, the design and analysis of explicit, strongly convergent methods for Stochastic DifferentialEquations with non-globally Lipschitz condition have become one of the current problems in stochastic numerical analysis. In this thesis, after preliminaries it is explained and numerically verified that under what conditions explicit Euler Maruyama method diverges. The tamed, projected and truncated Euler Maruyama methods which are strongly convergent under the relaxed conditions are introduced briefly and detailed strong convergence analysis of the tamed Euler-Maruyama Method is given. Additionally, the numerical performances of these methods are compared over some Stochastic Differential Equations.