Error Bounds for Small Jumps of Lévy Processes

Dia, El Hadj Aly

doi:10.1239/aap/1363354104

Cited by 7 publications

(2 citation statements)

References 17 publications

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“…On the other hand, according to Proposition 2.1 of Dia [9], we have a L q -upper bound of the error approximation in the one dimensional case for any real q > 0. This result on the strong error approximation remains valid for the multidimensional case.…”

Section: General Frameworkmentioning

confidence: 79%

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Importance sampling and statistical Romberg method for Lévy processes

Alaya

Hajji

Kebaier

2016

Stochastic Processes and their Applications

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An important family of stochastic processes arising in many areas of applied probability is the class of Lévy processes. Generally, such processes are not simulatable especially for those with infinite activity. In practice, it is common to approximate them by truncating the jumps at some cut-off size ε (ε ց 0). This procedure leads us to consider a simulatable compound Poisson process. This paper first introduces, for this setting, the statistical Romberg method to improve the complexity of the classical Monte Carlo one. Roughly speaking, we use many sample paths with a coarse cut-off ε β , β ∈ (0, 1), and few additional sample paths with a fine cut-off ε. Central limit theorems of LindebergFeller type for both Monte Carlo and statistical Romberg method for the inferred errors depending on the parameter ε are proved. This leads to an accurate description of the optimal choice of parameters with explicit limit variances. Afterwards, the authors propose a stochastic approximation method of finding the optimal measure change by Esscher transform for Lévy processes with Monte Carlo and statistical Romberg importance sampling variance reduction. Furthermore, we develop new adaptive Monte Carlo and statistical Romberg algorithms and prove the associated central limit theorems. Finally, numerical simulations are processed to illustrate the efficiency of the adaptive statistical Romberg method that reduces at the same time the variance and the computational effort associated to the effective computation of option prices when the underlying asset process follows an exponential pure jump CGMY model. MSC 2010: 60E07, 60G51, 60F05, 62L20, 65C05, 60H35.

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Section: General Frameworkmentioning

confidence: 79%

“…Moreover, under some regularity conditions on function F we can obtain an expansion of the weak error as in Proposition 2.2 and Remark 2.3 of [9]. So, it is worth to introduce the following assumption: there exist C F ∈ R and υ ε ց 0 as ε ց 0 such that…”

Section: General Frameworkmentioning

confidence: 99%

Importance sampling and statistical Romberg method for Lévy processes

Alaya

Hajji

Kebaier

2016

Stochastic Processes and their Applications

View full text Add to dashboard Cite

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Remarks on Lévy Process Simulation

Asmussen

2022

Advances in Modeling and Simulation

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Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

Cázares

Mijatović

2022

Finance Stoch

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We develop a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in Cázares and Mijatović (SBG approximation. GitHub repository. Available online at https://github.com/jorgeignaciogc/SBG.jl (2020)), exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes.

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Error Bounds for Small Jumps of Lévy Processes

Cited by 7 publications

References 17 publications

Importance sampling and statistical Romberg method for Lévy processes

Importance sampling and statistical Romberg method for Lévy processes

Remarks on Lévy Process Simulation

Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

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