Monte Carlo simulation for functionals of diffusion processes subject to exit probability

The focus of this thesis is to study various conditions under which the Euler scheme converges to the solution of Stochastic Differential Equations (SDE) under the assumption of Holder continuity in x subject to exit boundary conditions. We will implement Monte Carlo simulations to prove some order of convergence for Euler approximations of the SDE subject to exit boundary conditions when the drift and diffusion coefficients are not smooth. We resort to computer simulations using MATLAB to examine the convergence of the solution.