Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing

When approximating the expectation of a functional of a stochastic process, the efficiency and performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and reveal the available regularity, we consider cases in which analytic smoothing cannot be performed, and introduce a novel numerical smoothing approach by combining a root finding algorithm with one-dimensional integration with respect to a single well-selected variable. We prove that under appropriate conditions, the resulting function of the remaining variables is a highly smooth function, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, and our focus is on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we show the advantages of combining numerical smoothing with the ASGQ and QMC methods over ASGQ and QMC methods without smoothing and the Monte Carlo approach

Conditional Quasi-Monte Carlo with Constrained Active Subspaces

Conditional Monte Carlo or pre-integration is a useful tool for reducing variance and improving regularity of integrands when applying Monte Carlo and quasi-Monte Carlo (QMC) methods. To choose the variable to pre-integrate with, one need to consider both the variable importance and the tractability of the conditional expectation. For integrals over a Gaussian distribution, one can pre-integrate over any linear combination of variables. Liu and Owen (2022) propose to choose the linear combination based on an active subspace decomposition of the integrand. However, pre-integrating over such selected direction might be intractable. In this work, we address this issue by finding the active subspaces subject to the constraints such that pre-integration can be easily carried out. The proposed method is applied to some examples in derivative pricing under stochastic volatility models and is shown to outperform previous methods

Nonasymptotic Convergence Rate of Quasi-Monte Carlo: Applications to Linear Elliptic PDEs with Lognormal Coefficients and Importance Samplings

This study analyzes the nonasymptotic convergence behavior of the quasi-Monte Carlo (QMC) method with applications to linear elliptic partial differential equations (PDEs) with lognormal coefficients. Building upon the error analysis presented in (Owen, 2006), we derive a nonasymptotic convergence estimate depending on the specific integrands, the input dimensionality, and the finite number of samples used in the QMC quadrature. We discuss the effects of the variance and dimensionality of the input random variable. Then, we apply the QMC method with importance sampling (IS) to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient in bounded domains of $\mathbb{R}^d$, where the random coefficient is modeled as a stationary Gaussian random field parameterized by the trigonometric and wavelet-type basis. We propose two types of IS distributions, analyze their effects on the QMC convergence rate, and observe the improvements

Multilevel Path Branching for Digital Options

We propose a new Monte Carlo-based estimator for digital options with assets modelled by a stochastic differential equation (SDE). The new estimator is based on repeated path splitting and relies on the correlation of approximate paths of the underlying SDE that share parts of a Brownian path. Combining this new estimator with Multilevel Monte Carlo (MLMC) leads to an estimator with a complexity that is similar to the complexity of a MLMC estimator when applied to options with Lipschitz payoffs