A variance reduction technique using a quantized Brownian motion as a control variate

Lejay, Antoine; Reutenauer, Victor

doi:10.21314/jcf.2012.242

Cited by 10 publications

(4 citation statements)

References 24 publications

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“…Hence, a convenient variance reduction technique needs to be considered to deal with path-dependent functionals of Brownian motion. We mention for example variance reduction techniques in [41,39].…”

Section: Given the Vector Field¯mentioning

confidence: 99%

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A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations

Lejay

González

2020

Journal of Computational Physics

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A novel probabilistic scheme for solving the incompressible Navier-Stokes equations is studied, in which we approximate a generalized nonlinear Feyman-Kac formula. The velocity field is interpreted as the mean value of a stochastic process ruled by Forward-Backward Stochastic Differential Equations (FBSDEs) driven by Brownian motion. Following an approach by Delbaen, Qiu and Tang introduced in 2015, the pressure term is obtained from the velocity by solving a Poisson problem as computing the expectation of an integral functional associated to an extra BSDE. The FBSDEs components are numerically solved by using a forward-backward algorithm based on Euler type schemes for the local time integration and the quantization of the increments of Brownian motion following the algorithm proposed by Delarue and Menozzi in 2006. Numerical results are reported on spatially periodic analytic solutions of the Navier-Stokes equations for incompressible fluids. We illustrate the proposed algorithm on a two dimensional Taylor-Green vortex and three dimensional Beltrami flows.

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Section: Given the Vector Field¯mentioning

confidence: 99%

“…The order −1/2 −1/ is obtained for the Gaussian case. We refer to [55,41,19] for reduction variance techniques that use quantization.…”

Section: Estimation Of Conditional Expectationsmentioning

confidence: 99%

A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations

Lejay

González

2020

Journal of Computational Physics

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“…Functional quantization of Gaussian processes has become an active field of research in recent years since the seminal article Luschgy and Pagès (2002). As far as applications are concerned, cubature methods (Pagès and Printems 2005a;Corlay 2010b) and variance reduction methods (Corlay and Pagès 2010;Lejay and Reutenauer 2008) based on functional quantization have been proposed. However, as the numerical use of functional quantizers requires the evaluation of the Karhunen-Loève eigenfunctions, this method was restricted to processes for which a closed-form expression for this expansion is known, such as Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Multifractional Stochastic Volatility Models

Corlay

Lebovits

Véhel

2013

Mathematical Finance

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The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.

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“…The emphasis is on the functional quantization of Gaussian processes. Section 2 briefly covers the first functional quantization-based variance reduction method that was proposed in [25,16]. Section 3 outlines the links between quantization and stratification with an emphasis on the Gaussian case.…”

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confidence: 99%

Functional quantization-based stratified sampling methods

Corlay

Pagès

2015

Monte Carlo Methods and Applications

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In this article, we propose several quantization-based strati ed sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of strati cation lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with strati ed sampling. We rst put the emphasis on the consistency of quantization for partitioning the state space in strati ed sampling methods in both nite and in nite-dimensional cases. We show that the proposed quantization-based strata design has uniform e ciency among the class of Lipschitz continuous functionals. Then a strati ed sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multifactor di usions. 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.

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A variance reduction technique using a quantized Brownian motion as a control variate

Cited by 10 publications

References 24 publications

A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations

A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations

Multifractional Stochastic Volatility Models

Functional quantization-based stratified sampling methods

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