Multilevel Quasi-Monte Carlo Methods With Applications In Finance And Statistics

Quasi-Monte Carlo（QMC）method is a kind of numerical simulation tool which can give higher convergence rate than Monte Carlo（MC）method.Multilevel Monte Carlo（MLMC）method can also reduce the complexity of MC method which has wide application scope.The advantages of QMC and MLMC methods inspire us to explore how to combine QMC and MLMC to achieve further improvement in financial and statistical problems.In this thesis,we focus on two problems and investigate how to deal with the problems with the help of QMC and MLMC methods.The first problem is about risk management in finance area.We consider the problem of estimating the probability of a large loss from a financial portfolio,where the future loss is expressed as a conditional expectation.Since the conditional expectation is usually intractable in most cases,nested simulation is used to handle the problem.To reduce the complexity of nested simulation,we present a method that combines MLMC and QMC.In the outer simulation,we use MC to generate financial scenarios.In the inner simulation,we use QMC to estimate portfolio loss in each scenarios.We prove that using QMC can accelerate the convergence rates in both the crude nested simulation and the multilevel nested simulation.Under certain conditions,the complexity of MLMC can be reduced to(^-2（log）²)by incorporating QMC with MSE of（²）.On the other hand,we find that MLMC encounters a high-kurtosis phenomenon due to the existence of indicator functions.To remedy this,we propose a smoothed MLMC method which uses logistic sigmoid functions to approximate the indicator functions.Numerical results show that the optimal complexity(^-2)is almost attained when using QMC methods in both MLMC and smoothed MLMC,even in moderate high dimensions.The second problem we are concerned with is about variational Bayes（VB）method for Bayesian inference in statistical modeling.Some VB algorithms are proposed to han-dle intractable likelihoods with applications such as approximate Bayesian computation problems.We propose unbiased score function（SF）and re-parameterization（RP）estima-tors based on unbiased MLMC method for the gradient of Kullback-Leibler divergence between the posterior distribution and the variational distribution when the likelihood is intractable,but can be estimated unbiasedly.The new VB algorithm differs from the VB algorithms in the literature which usually render biased gradient estimators.Moreover,we incorporate randomized QMC（RQMC）sampling within the MLMC-based gradient estimators,which provides a favorable rate of convergence in numerical integration.The-oretical results are provided to make sure RQMC methods are feasible in the new settings,which generalize the results under MC scheme.Numerical experiments show that using RQMC in MLMC greatly speeds up the VB algorithm,and finds a better approximation to the posterior distribution than some existing competitors do.