Multilevel path simulation for weak approximation schemes with application to Lévy-driven SDEs

Belomestny, Denis; Nagapetyan, Tigran

doi:10.3150/15-bej764

Cited by 8 publications

(16 citation statements)

References 23 publications

(28 reference statements)

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“…is satisfied, is studied in Belomestny and Nagapetyan [2]. Recall that (15) is violated in our construction, except in trivial cases, see (6).…”

Section: A Random Bit Multilevel Algorithm In This Section We Considmentioning

confidence: 99%

“…, 2 q }, yields a much larger number of pairwise independent random variables with the same uniform distribution. The number of random bits used by the algorithm A Bbit ε is of the order ε −2 • (ln(ε −1 )) 5/2 , see the proof of Theorem 8, and it can be reduced further to ε −2 • (ln(ε −1 )) 2 • ln(ln(ε −1 )), as outlined in Remark 9. Since the upper bound in (1) is sharp, the number of random bits is asymptotically negligible compared to the overall cost of A Bbit ε .…”

Section: Introductionmentioning

confidence: 99%

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Random bit multilevel algorithms for stochastic differential equations

Giles

Hefter

Mayer

et al. 2019

Journal of Complexity

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We study the approximation of expectations E(f (X)) for solutions X of SDEs and functionals f : C([0, 1], R r ) → R by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals f from the Lipschitz class w.r.t. the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms.

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“…is satisfied, is studied in Belomestny and Nagapetyan [2]. Recall that (15) is violated in our construction, except in trivial cases, see (6).…”

Section: A Random Bit Multilevel Algorithm In This Section We Considmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Random bit multilevel algorithms for stochastic differential equations

Giles

Hefter

Mayer

et al. 2019

Journal of Complexity

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“…In [1] Belomestny and Nagapetyan introduced the Weak MLMC method, proving that MLMC based on the weak Euler scheme maintains C = O(ε −2 (log ε) 2 ) when doing the right coupling of levels. Concretely, for an arbitrary level l, let ξ f l,i and ξ c l,i , i = 1, .…”

Section: Weak Mlmcmentioning

confidence: 99%

“…• the Euler MLMC method with binomial random variables introduced by Belomestny and Nagapetyan [1] (Euler),…”

Section: Numerical Examplementioning

confidence: 99%

Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes

Debrabant

Ghasemifard

Mattsson

2019

Applied Mathematics Letters

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In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

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“…, m L−1 }. Maybe these restrictions could be relaxed when replacing the Brownian increments in the Euler schemes by Rademacher random variables like in the weak MLMC method introduced by Belomestny and Nagapetyan [2]. Nonetheless the derivation of concentration bounds for the weak MLMC estimators would require a different approach.…”

Section: Introductionmentioning

confidence: 99%

Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

Jourdain¹,

Kebaier²

2019

Electron. J. Probab.

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In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

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Multilevel path simulation for weak approximation schemes with application to Lévy-driven SDEs

Cited by 8 publications

References 23 publications

Random bit multilevel algorithms for stochastic differential equations

Random bit multilevel algorithms for stochastic differential equations

Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes

Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

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