| This dissertation presents some theoretical and application results on Monte Carlo and Quasi-Monte Carlo.; In Chapter 1, we give a brief introduction.; In Chapter 2, we present an extension of a known result on ( t, m, s)-nets/(t, s)-sequences—the best-known (so far) low discrepancy point sets. The same chapter also presents the extension of the range of some parameters.; In Chapter 3, we present a theorem on lattice rules of intermediate rank. Based on this theorem, upper bounds for multiple integration error can be derived, and the existence of “good” lattice rules of intermediate rank with general order is guaranteed. This theorem recovers known results on lattice rules of rank-1 and maximal rank.; In Chapter 4, we study the applications of Monte Carlo and Quasi-Monte Carlo methods in transport problems, the geometric/exponential convergence by adaptive Monte Carlo methods—Sequential Correlated Sampling Method and Adaptive Importance Sampling Method. For Sequential Correlated Sampling Method, we prove a theorem providing geometric/exponential convergence. For Adaptive Importance Sampling Method, we provide numerical evidences of geometric convergence in both discrete and continuous cases.; In Chapter 5, we demonstrate the applications of Monte Carlo and Quasi-Monte Carlo in finance—option pricing. Three methods involving Monte Carlo simulation will be addressed.; In Chapter 6, we give a brief summary of this dissertation.; The results in this dissertation can be applied to problems involving (high dimensional) integrals, (large-scale) matrix equations and other problems where the traditional methods are hard, to obtain reasonable results. Such problems arise in many areas, e.g., particle transport problems in nuclear engineering, operations research, statistics, scientific computation and financial engineering—derivative pricing. |
