scholarly journals Functional quantization-based stratified sampling methods

2015 ◽  
Vol 21 (1) ◽  
pp. 1-32 ◽  
Author(s):  
Sylvain Corlay ◽  
Gilles Pagès
Keyword(s):  
Variance Reduction ◽  
Sampling Methods ◽  
Brownian Bridge ◽  
Reduction Factor ◽  
Stratified Sampling ◽  
Lipschitz Continuous ◽  
Infinite Dimensional ◽  
Path Dependent ◽  
Functional Quantization ◽  
Continuous Functionals

AbstractIn this article, we propose several quantization-based stratified sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of stratification lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with stratified sampling. We first put the emphasis on the consistency of quantization for partitioning the state space in stratified sampling methods in both finite and infinite-dimensional cases. We show that the proposed quantization-based strata design has uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein–Uhlenbeck processes. We derive in detail the case of Ornstein–Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction factor.

The use of variance reduction, relative error and bias in testing the performance of M/G/1 retrial queues estimators in Monte Carlo simulation

2018 ◽  
Vol 24 (3) ◽  
pp. 165-178 ◽  
Author(s):  
Kenza Tamiti ◽  
Megdouda Ourbih-Tari ◽  
Abdelouhab Aloui ◽  
Khelidja Idjis
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Relative Error ◽  
Performance Measures ◽  
Random Sampling ◽  
Variance Reduction ◽  
Sampling Methods ◽  
Reduction Factor ◽  
Retrial Queues ◽  
C Compiler

Abstract This paper deals with Monte Carlo simulation and focuses on the use of the concepts of variance reduction, relative error and bias in testing the performance of stationary M/G/1 retrial queues estimators using either Random Sampling (RS) or Refined Descriptive Sampling (RDS) to generate input samples. For this purpose, a software under Linux system using the C compiler was designed and realized providing the performance measures of such system and the statistical concepts of bias, relative error and accuracy using both sampling methods. As a conclusion, it has been shown that the performance of stationary M/G/1 retrial queues estimators is better using RDS than RS and sometimes by a substantial variance reduction factor.

scholarly journals Some hemivariational inequalities in the Euclidean space

2019 ◽  
Vol 9 (1) ◽  
pp. 958-977 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Hemivariational Inequalities ◽  
Existence Of Weak Solutions ◽  
Lipschitz Continuous ◽  
Variational Structure ◽  
Locally Lipschitz ◽  
Sign Changing Solutions ◽  
Continuous Functionals

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

scholarly journals CONVENIENT MULTIPLE DIRECTIONS OF STRATIFICATION

2011 ◽  
Vol 14 (06) ◽  
pp. 867-897 ◽  
Author(s):  
BENJAMIN JOURDAIN ◽  
BERNARD LAPEYRE ◽  
PIERGIACOMO SABINO
Keyword(s):  
Variance Reduction ◽  
Optimal Allocation ◽  
Computational Cost ◽  
Principal Component ◽  
Latin Hypercube Sampling ◽  
Allocation Rule ◽  
Black Scholes ◽  
Normal Vectors ◽  
Computationally Intensive ◽  
Path Dependent

This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along non-orthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.

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