A framework for adaptive Monte Carlo procedures

Lapeyre, Bernard; Lelong, Jérôme

doi:10.1515/mcma.2011.002

Cited by 22 publications

(23 citation statements)

References 23 publications

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“…At the beginning, note that for n ∈ N the existence of θ * n is ensured by Proposition 2.2. Concerning, the first assertion, we have to check both assumptions of Theorem 3.1 in [16]. The first one given by…”

Section: Constrained Stochastic Algorithmmentioning

confidence: 99%

“…The first one, based on a truncation procedure called "Projection à la Chen", is introduced by Chen in [6,7] and investigated later by several authors (see, e.g., Andrieu, Moulines and Priouret in [1] and Lelong in [17]). 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].…”

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: Constrained Stochastic Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Importance sampling and statistical Romberg method

Alaya¹,

Hajji²,

Kebaier³

2015

Bernoulli

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“…Quasi-Monte Carlo versions of these two algorithms have been introduced in [21]. More recently, adaptive approaches have been developed for stratified sampling [5,8] and for importance sampling [11]. The idea of adaptive algorithms is to make use of past simulations to help better leading future ones.…”

Section: Introductionmentioning

confidence: 99%

Robust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options

Luigi

Lelong

Maire

2016

Applied Numerical Mathematics

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International audienceWe improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of multidimensional vanilla options in the Black–Scholes framework which emphasizes the numerical problems of integrating non-smooth functions. In high dimensions, this new algorithm is used as a control variate after a dimension reduction based on principal component analysis. Numerical tests are performed on the Genz package, on the pricing of basket, put on minimum and digital options in dimensions up to ten

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“…In practice, the perfect probability measure is generally unavailable, and the importance sampling problem can be formulated as a parametric optimization problem. Based on Monte Carlo sampling techniques, Su and Fu [12] used stochastic approximation (SA) to solve the resulting stochastic optimization problem in the geometric Brownian motion setting (also refer to Lapeyre and Lelong [13]), and Kawai [14] and Kawai [8] extended the approach to Lévy processes.…”

Section: Introductionmentioning

confidence: 99%

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

Jiang

2016

Operations Research Letters

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We formulate the problem of determining the optimal importance sampling measure change for pricing financial derivatives under Lévy processes as a parametric optimization problem, and propose a solution approach using sample average approximation (SAA) with Newton iteration to find the optimal parameters in the Esscher probability measure change. Theoretical results, such as convergence rate of the optimal solutions, are provided. A numerical example illustrates the effectiveness of the approach.

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A framework for adaptive Monte Carlo procedures

Cited by 22 publications

References 23 publications

Importance sampling and statistical Romberg method

Importance sampling and statistical Romberg method

Robust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

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