Unconstrained recursive importance sampling

Lemaire, Vincent; Pagès, Gilles

doi:10.1214/09-aap650

Cited by 41 publications

(68 citation statements)

References 22 publications

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“…In their recent paper [18], Lemaire and Pagès proposed a new procedure using Robbins-Monro algorithm that satisfies the classical nonexplosion condition (NEC). In fact, a new expression of the gradient is obtained by a third change of probability.…”

Section: Unconstrained Stochastic Algorithmmentioning

confidence: 99%

“…The use of this procedure in the context of importance sampling is initially proposed by Arouna in [2] and investigated afterward by Lapeyre and Lelong in [16]. The second alternative, is more recent and introduced by Lemaire and Pagès in [18]. In fact, they proposed an unconstrained procedure by using extensively the regularity of the involved density and they prove the convergence of this algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Importance sampling and statistical Romberg method

Alaya¹,

Hajji²,

Kebaier³

2015

Bernoulli

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The efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods. Among these methods, we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter θ. The optimal choice of θ is approximated using Robbins-Monro procedures, provided that a nonexplosion condition is satisfied. Otherwise, one can use either a constrained Robbins-Monro algorithm (see, e.g., Arouna (Monte Carlo Methods Appl. 10 (2004) 1-24) and Lelong (Statist. Probab. Lett. 78 (2008) 2632-2636)) or a more astute procedure based on an unconstrained approach recently introduced by Lemaire and Pagès in (Ann. Appl. Probab. 20 (2010) 1029-1067). In this article, we develop a new algorithm based on a combination of the statistical Romberg method and the importance sampling technique. The statistical Romberg method introduced by Kebaier in (Ann. Appl. Probab. 15 (2005) 2681-2705) is known for reducing efficiently the complexity compared to the classical Monte Carlo one. In the setting of discritized diffusions, we prove the almost sure convergence of the constrained and unconstrained versions of the Robbins-Monro routine, towards the optimal shift θ * that minimizes the variance associated to the statistical Romberg method. Then, we prove a central limit theorem for the new algorithm that we called adaptive statistical Romberg method. Finally, we illustrate by numerical simulation the efficiency of our method through applications in option pricing for the Heston model.Eψ(X T ) = Eg(θ, X T ).

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Section: Unconstrained Stochastic Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Importance sampling and statistical Romberg method

Alaya¹,

Hajji²,

Kebaier³

2015

Bernoulli

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“…In Frikha (), two variance reduction techniques have been developed in order to reduce the asymptotic variance in the CLT (3.12). The first one is based on the unconstrained IS stochastic algorithm originally developed in Lemaire and Pagès () and then applied to both VaR and CVaR in Bardou et al. ().…”

Section: Computational and Numerical Aspects Of Cvar Hedgingmentioning

confidence: 99%

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

Bardou

Frikha

Pagès

2013

Mathematical Finance

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In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.

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“…To name a few examples and references, it has applications in numerical integration [32,22,25] and in rare event simulation [9,31,7]. The idea is to direct the simulations 1 to important regions of space through an appropriate change of measure.…”

Section: Introductionmentioning

confidence: 99%

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Gobet

Turkedjiev²

2017

Stochastic Processes and their Applications

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Please cite this article as: E. Gobet, P. Turkedjiev, Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations, Stochastic Processes and their Applications (2016), http://dx.doi.org/10. 1016/j.spa.2016.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares MonteCarlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

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Unconstrained recursive importance sampling

Cited by 41 publications

References 22 publications

Importance sampling and statistical Romberg method

Importance sampling and statistical Romberg method

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

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