Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions

Lelong, Jérôme

doi:10.1016/j.spl.2008.02.034

Cited by 18 publications

(24 citation statements)

References 8 publications

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“…So, (17) and (18) tell us A(θ ) ≤ c 2 c 3 n p , and we deduce the existence of a deterministic constant c 4 depending only on p, η, T , b, σ and ψ such that…”

Section: Unconstrained Stochastic Algorithmmentioning

confidence: 52%

“…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].…”

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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“…So, (17) and (18) tell us A(θ ) ≤ c 2 c 3 n p , and we deduce the existence of a deterministic constant c 4 depending only on p, η, T , b, σ and ψ such that…”

Section: Unconstrained Stochastic Algorithmmentioning

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Importance sampling and statistical Romberg method

Alaya¹,

Hajji²,

Kebaier³

2015

Bernoulli

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“…The new idea of using one importance sampling parameter per level was later taken up in [6] but coupled with stochastic approximation to build adaptive estimators. Actually, minimizing λ −→ σ 2 (λ) can be achieved by using the randomly truncated Robbins Monro algorithm proposed by Chen et al [7,8] and later investigated in the context of importance sampling by Lapeyre and Lelong [25] and Lelong [26]. The numerical stability of these stochastic algorithms strongly depends on the choice of the descent step -often referred to as the gain sequence -which proves to be highly sensitive in practice.…”

Section: 2)mentioning

confidence: 99%

Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Kebaier

Lelong

2017

Methodol Comput Appl Probab

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In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to handle the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

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“…Note that the idea of truncations goes back to Khasminskii and Nevelson (1972) and Fabian (1978) (see also Chen and Zhu (1986), Chen et al(1987), Andradóttir (1995), Sharia (1997), Tadic (1997Tadic ( ,1998, Lelong (2008). A comprehensive bibliography and some comparisons can be found in Sharia (2014)).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of truncated stochastic approximation procedures

Sharia

Zhong

2017

Math. Meth. Stat.

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We study asymptotic behaviour of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quite mild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable form of the central limit theorem can be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.

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Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions

Cited by 18 publications

References 8 publications

Importance sampling and statistical Romberg method

Importance sampling and statistical Romberg method

Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Asymptotic behavior of truncated stochastic approximation procedures

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