This research's purpose is to optimize an existing method to simulate stochas- tic integrals using Monte Carlo when the cost of function... Show moreThis research's purpose is to optimize an existing method to simulate stochas- tic integrals using Monte Carlo when the cost of function evaluation is dimension dependent. In the area of mathematical nance, we often need to price a path- dependent nancial derivative. This will result in the computation of E[g(B( ))], where g stands for a payoff function, and B is the Brownian Motion. A simple way to approximate this expectation is to take the average of the functional over a large num- ber of sample paths. Each path is approximated by a d-dimensional random vector. A larger d will provide a more accurate result. However, due to the limitation in time cost and computer memory, some large dimensions are not easy to be implemented. Therefore, we introduce the multi-level technique that is based on multi-grid ideas. It can be used to reduce the computational complexity for these kind of problems. Moreover, when we apply the multi-level technique, the proper sample size for each subspace integration needs to be computed in order to satisfy our guaranteed conser- vative xed width con dence intervals. Thus, the automatic sample size algorithm (two stage con dence interval algorithm) is used in conjunction with the multi-level method. M.S. in Applied Mathematics, December 2013 Show less