Applying Perfect Simulation to solve stochastic difference equations that arise from certain time series models

The CTAR and ARCH models are nonlinear time series models used to model financial data. These models are represented by nonlinear stochastic equations. This thesis uses Perfect Simulation techniques to provide algorithms to find solutions to these equations and other problems originated from the CTAR(1) and ARCH(1) models. The discretization of the CTAR(1) model gives rise to a nonlinear stochastic difference equation (SDE). The first part of this thesis provides a perfect simulation algorithm (using the Slicing Coupler) to find the stationary distribution of the solution of such S DE. The second part of this thesis provides a perfect simulation algorithm (using the Harris Coupler) to find the stationary distribution of the solution of the ARCH(1) model. The final part of this thesis provides a perfect simulation algorithm (using Perfect IMH) for a Bayesian estimation problem, allowing us to sample directly from the posterior distribution of the parameters of the ARCH(1) model.