Chapter 10 A Hilbert Space Approach to Variance Reduction

Szechtman, Roberto

doi:10.1016/s0927-0507(06)13010-8

Cited by 6 publications

(4 citation statements)

References 42 publications

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“…As a result its variance cannot be further reduced using the controls. For a Hilbert space-based exposition of the CV technique, derivations, and further results, see, e.g., (Szechtman 2006).…”

Section: A Unified Representation Of Sensitivity Estimatorsmentioning

confidence: 99%

Control variates for sensitivity estimation

Borogovac

Sun

Vakili

2010

Proceedings of the 2010 Winter Simulation Conference

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We adapt a newly proposed generic approach to control variate selection to the problem of efficient estimation of sensitivity of financial security prices to model parameters, the so-called Greeks. We show that estimators based on pathwise and likelihood ratio methods can be cast in a general setting where generic control variates can be systematically defined for their estimation. In general, the means of such controls cannot be exactly calculated. One can use the Biased or Estimated Control Variates approach and estimate the means via simulation, or use the approach of DataBase Monte Carlo (DBMC) which also requires estimation of control means via simulation. We consider a parametric setting where price sensitivities need to be estimated repeatedly at multiple parameters. The fact that the same controls can be used for multiple estimation problems can justify the setup cost. The approach is illustrated via simple examples and preliminary computational results are provided.

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Section: A Unified Representation Of Sensitivity Estimatorsmentioning

confidence: 99%

Control variates for sensitivity estimation

Borogovac

Sun

Vakili

2010

Proceedings of the 2010 Winter Simulation Conference

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“…The term β * (X − µ) corresponds to the familiar orthogonal projection of Y on the linear subspace spanned by X 1 , • • • , X k in the Hilbert space setting. Z is the innovation term corresponding to the variation/information in Y that is not explainable by the control variates (see, e.g., (Szechtman 2006)).…”

Section: Classical Control Variate (Cv)mentioning

confidence: 99%

“…Once the source of information, i.e., the set of controls, is identified, the mechanisms for optimal information extraction and transfer are well understood and analyzed (See, e.g., (Glasserman 2004), (Asmussen and Glynn 2007), (Nelson 1990), (Szechtman 2003), (Szechtman 2006), (Lavenberg and Welch 1981)). On the other hand, while there are some guidelines and common approaches for identifying/discovering controls, the identification/discovery process is fairly ad-hoc and its success depends critically on the ingenuity of the user.…”

Section: Introductionmentioning

confidence: 99%

Control variate technique: A constructive approach

Borogovac¹,

Vakili²

2008

2008 Winter Simulation Conference

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The technique of control variates requires that the user identify a set of variates that are correlated with the estimation variable and whose means are known to the user. We relax the known mean requirement and instead assume the means are to be estimated. We argue that this strategy can be beneficial in parametric studies, analyze the properties of controlled estimators, and propose a class of generic and effective controls in a parametric estimation setting. We discuss the effectiveness of the estimators via analysis and simulation experiments.

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“…A nice review of control variates and their use from alternative viewpoints is provided by Glynn & Szechtman (2002) and Szechtman (2006). They present control variates under the Hilbert space viewpoint, as well as their relationships with numerical integration, antithetics, stratification and rotation sampling.…”

Section: Related References For the Use Of Control Variates For Variance Reduction In Monte Carlomentioning

confidence: 99%

On variance reduction for Markov chain Monte Carlo

Tsourti¹,

Τσούρτη²

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In the present thesis we are concerned with appropriate variance reduction methods for specific classes of Markov Chain Monte Carlo (MCMC) algorithms. The variance reduction method of main interest here is that of control variates. More particularly, we focus on control variates of the form U = G − P G, for arbitrary function G, where P G stands for the one-step ahead conditional expectation, that have been proposed by Henderson (1997). A key issue for the efficient implementation of control variates is the appropriate estimation of corresponding coefficients. In the case of Markov chains, this involves the solution of Poisson equation for the function of initial interest, which in most cases is intractable. Dellaportas & Kontoyiannis (2012) have further elaborated on this issue and they have proven optimal results for the case of reversible Markov chains, avoiding that function. In this context, we concentrate on the implementation of those results for MetropolisHastings (MH) algorithm, a popular MCMC technique. In the case of MH, the main issue of concern is the assessment of one-step ahead conditional expectations, since these are not usually available in closed form expressions. The main contribution of this thesis is the development and evaluation of appropriate techniques for dealing with the use of the above type of control variates in the MH setting. The basic approach suggested is the use of Monte Carlo method for estimating one-step ahead conditional expectations as empirical means. In the case of MH this is a straightforward task requiring minimum additional analytical effort. However, it is rather computationally demanding and, hence, alternative methods are also suggested. These include importance sampling of the available data resulting from the algorithm (that is, the initially proposed or finally accepted values), additional application of the notion of control variates for the estimation of P G’s, or parallel exploitation of the values that are produced in the frame of an MH algorithm but not included in the resulting Markov chain (hybrid strategy). The ultimate purpose is the establishment of a purely efficient strategy, that is, a strategy where the variance reduction attained overcomes the additional computational cost imposed. The applicability and efficiency of the methods is illustrated through a series of diverse applications.

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Chapter 10 A Hilbert Space Approach to Variance Reduction

Cited by 6 publications

References 42 publications

Control variates for sensitivity estimation

Control variates for sensitivity estimation

Control variate technique: A constructive approach

On variance reduction for Markov chain Monte Carlo

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