Improving Approximate Bayesian Computation via Quasi-Monte Carlo

ABC (approximate Bayesian computation) is a general approach for dealing with models with an intractable likelihood. In this work, we derive ABC algorithms based on QMC (quasi-Monte Carlo) sequences. We show that the resulting ABC estimates have a lower variance than their Monte Carlo counter-parts. We also develop QMC variants of sequential ABC algorithms, which progressively adapt the proposal distribution and the acceptance threshold. We illustrate our QMC approach through several examples taken from the ABC literature.     

拟蒙特卡罗法在亚洲期权定价中的应用

亚洲期权是场外交易中几种最受欢迎的新型期权之一,但它的价格却没有解析表达式,到目前为止,亚洲期权的定价仍是个公开问题.本文采用拟蒙特卡罗法中的Halton序列来估计它的价格,数值结果表明当观察点的个数N13时,它比蒙特卡罗法要好.本文还利用MATLAB程序生成了随机Halton序列,并将它与控制变量法结合起来估计亚洲期权的价格,估计值标准差的比较表明它在大多情况下比相应的蒙特卡罗法的估计效果要好.     

Quasi-Monte Carlo integration in unanchored Sobolev spaces

We find a concrete sequence of N points, for which the squared worst-case quasi-Monte Carlo error in the Hilbert space of continuous functions defined on [0, 1] with square integrable second derivative is smaller than for the centered regular lattice point set.     

Quasi-Monte Carlo methods for two-stage stochastic mixed-integer programs

Mathematical Programming - We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second...     

Quasi-Monte Carlo for Highly Structured Generalised Response Models

Highly structured generalised response models, such as generalised linear mixed models and generalised linear models for time series regression, have become an indispensable vehicle for data analysis and inference in many areas of application. However, their use in practice is hindered by high-dimensional intractable integrals. Quasi-Monte Carlo (QMC) is a dynamic research area in the general problem of high-dimensional numerical integration, although its potential for statistical applications is yet to be fully explored. We survey recent research in QMC, particularly lattice rules, and report on its application to highly structured generalised response models. New challenges for QMC are identified and new methodologies are developed. QMC methods are seen to provide significant improvements compared with ordinary Monte Carlo methods.      

拟蒙特卡罗模拟方法在金融计算中的应用研究

在本文中我们展示了低差异序列的一些特点,利用拟蒙特卡罗模拟方法中的Halton、Faure、Sobol序列来对期权进行数值定价分析,数值实验结果表明:在低维数的条件下Hal- ton、Faure、Sobol序列比(伪)蒙特卡罗模拟方法好,在高维数的条件下,Halton序列比较敏感,Faure、Sobol序列比其它方法表现好.     

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers and contrasts the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first-order QMC rules versus higher-order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.     

拟蒙特卡罗方法用于估计多元回归函数

拟蒙特卡罗（QMC）方法被广泛用于解决数值分析和统计学中的各种问题，比如数值积分，最优化，试验设计，随机过程的模拟等．本文研究该方法在估计多元回归函数中的应用．证明了，在相当一般的条件下，均匀设计（或者，“代表点设计”）与回归函数傅里叶系数的QMC估计（对应地，使用拟随机重要性抽样的QMC估计）一起，构成一个回归函数的渐近最优投影估计．     

Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulations. We assume a Black–Scholes market with time-dependent volatilities, and we compute the deltas by means of the Malliavin Calculus as an extension of the procedures employed by Kohatsu-Higa and Montero (Physica A 320:548–570, 2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. To face this challenge we then introduce a new and faster Cholesky algorithm for block matrices that makes the Linear Transformation more convenient. We also propose a new-path generation technique based on a Kronecker Product Approximation. Our procedure shows the same accuracy as the Linear Transformation used for the computation of deltas and prices in the case of correlated asset returns, while requiring a shorter computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004a, Risk 17(5):97–101, b).     

A Constructive Approach to Strong Tractability Using Quasi-Monte Carlo Algorithms

We prove in a constructive way that multivariate integration in appropriate weighted Sobolev classes is strongly tractable and the ε-exponent of strong tractability is 1 (which is the best-possible value) under a stronger assumption than Sloan and Woźniakowski's assumption. We show that quasi-Monte Carlo algorithms based on the Sobol sequence and Halton sequence achieve the convergence order O(n^−1+δ) for any δ>0 independent of the dimension with a worst-case deterministic guarantee (where n is the number of function evaluations). This implies that quasi-Monte Carlo algorithms based on the Sobol and Halton sequences converge faster and therefore are superior to Monte Carlo methods independent of the dimension for integrands in suitable weighted Sobolev classes.     

Quasi-Monte Carlo methods for numerical integration of multivariate Haar series II

The present paper contains a comparison of different classes of multivariate Haar series that have been studied with respect to numerical integration, new properties ofE _s ^α -classes and numerical results. Research supported by the Austrian Science Foundation (FWF), project no. P11143-MAT.     

Quasi-Monte Carlo methods for the numerical integration of multivariate walsh series

In [1], a method for the numerical integration of multivariate Walsh series, based on low-discrepancy point sets, was developed. In the present paper, we improve and generalize error estimates given in [1] and disprove a conjecture stated in [1,2].     

美式障碍期权定价的总体最小二乘拟蒙特卡罗模拟方法

障碍期权的价格依赖于其标的资产的价格路径,实际市场中标的资产的价格变化存在跳跃现象。本文在跳跃扩散模型下使用总体最小二乘拟蒙特卡罗方法(TLSFM)对美式障碍期权定价问题进行了研究。TLSFM使用随机化的Faure序列并结合总体最小二乘回归方法,改进了Longstaff等提出的最小二乘蒙特卡罗模拟方法(LSM)。通过基于TLSFM与LSM和改进的三叉树方法的美式障碍期权定价结果的比较分析,说明了基于TLSFM的美式障碍期权定价具有结果稳定,时效性更强的优势。     

Improved Diffusion Monte Carlo

We propose a modification, based on the RESTART (repetitive simulation trials after reaching thresholds) and DPR (dynamics probability redistribution) rare event simulation algorithms, of the standard diffusion Monte Carlo (DMC) algorithm. The new algorithm has a lower variance per workload, regardless of the regime considered. In particular, it makes it feasible to use DMC in situations where the “naïve” generalization of the standard algorithm would be impractical due to an exponential explosion of its variance. We numerically demonstrate the effectiveness of the new algorithm on a standard rare event simulation problem (probability of an unlikely transition in a Lennard‐Jones cluster), as well as a high‐frequency data assimilation problem. © 2014 Wiley Periodicals, Inc.     

Quasi-Monte Carlo Sampling for Solving Partial Differential Equations by Deep Neural Networks

Solving partial differential equations in high dimensions by deep neural networks has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo method, together with a deep neural network, is used to overcome the curse of dimensionality, while classical methods fail. Often, a neural network outperforms classical numerical methods in terms of both accuracy and efficiency. In this paper, we propose to use quasi-Monte Carlo sampling, instead of Monte-Carlo method to approximate the loss function. To demonstrate the idea, we conduct numerical experiments in the framework of deep Ritz method. For the same accuracy requirement, it is observed that quasi-Monte Carlo sampling reduces the size of training data set by more than two orders of magnitude compared to that of Monte-Carlo method. Under some assumptions, we can prove that quasi-Monte Carlo sampling together with the deep neural network generates a convergent series with rate proportional to the approximation accuracy of quasi-Monte Carlo method for numerical integration. Numerically the fitted convergence rate is a bit smaller, but the proposed approach always outperforms Monte Carlo method.