| Propp and Wilson (1996) proposed the idea of coupling from the past (CFTP) for the purpose of simulating exactly from the limiting distribution of a Markov chain. The multishift coupler (Wilson, 2000b) is an important tool to apply the idea of CFTP on a continuous state space. In this thesis, we describe a coupled Euler scheme which applies the multishift coupler to the usual Euler method of simulation of a stochastic differential equation (SDE) at fixed time. This supplies a technique to prove that the usual Euler approximation can achieve any pre-specified total variation distance from the exact distribution of the solution to a constant diffusion SDE. As for non-constant diffusion SDEs, we can apply the Ito formula to transform them into constant diffusion SDEs. In addition, we describe ideas for extending our algorithm to other time periods and propose a possible way of obtaining perfect samples. |
