Hostname: page-component-f554764f5-wjqwx Total loading time: 0 Render date: 2025-04-23T05:12:10.013Z Has data issue: false hasContentIssue false
  • English
  • Français

On the rate of convergence for an α-stable central limit theorem under sublinear expectation

Part of: Distribution theory - Probability Partial differential equations, initial value and time-dependent initial-boundary value problems Limit theorems

Published online by Cambridge University Press:  03 October 2024

Mingshang Hu ,
Lianzi Jiang  and
Gechun Liang Open the ORCID record for Gechun Liang [Opens in a new window]
Mingshang Hu*
Affiliation:
Shandong University
Lianzi Jiang*
Affiliation:
Shandong University of Science and Technology
Gechun Liang*
Affiliation:
The University of Warwick
*
*Postal address: Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China. Email: humingshang@sdu.edu.cn
**Postal address: College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, PR China. Email: jianglianzi95@163.com
***Postal address: Department of Statistics, The University of Warwick, Coventry CV4 7AL, UK. Email: g.liang@warwick.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations which characterize nonlinear $\alpha$-stable Lévy processes under a sublinear expectation space with $\alpha\in(1,2)$. We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry–Esseen bound and convergence rate for the $\alpha$-stable central limit theorem under sublinear expectation.

Keywords

Stable central limit theoremconvergence ratesublinear expectationmonotone scheme methodpartial integro-differential equation

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E07: Infinitely divisible distributions; stable distributions 65M15: Error bounds
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 209 - 233
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arras, B., Mijoule, G., Poly, G. and Swan, Y. (2017). A new approach to the Stein–Tikhomirov method: With applications to the second Wiener chaos and Dickman convergence. Preprint, arXiv:1605.06819.Google Scholar
Barles, G. and Jakobsen, E. R. (2002). On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations. M2AN Math. Model. Numer. Anal. 36, 33–54.CrossRefGoogle Scholar
Barles, G. and Jakobsen, E. R. (2005). Error bounds for monotone approximation schemes for Hamilton–Jacobi–Bellman equations. SIAM J. Numer. Anal. 43, 540–558.CrossRefGoogle Scholar
Barles, G. and Jakobsen, E. R. (2007). Error bounds for monotone approximation schemes for parabolic Hamilton–Jacobi–Bellman equations. Math. Comput. 76, 1861–1893.CrossRefGoogle Scholar
Barles, G. and Souganidis, P. E. (1991). Convergence of approximation schemes for fully nonlinear second order equations. Asymp. Anal. 4, 271–283.Google Scholar
Bayraktar, E. and Munk, A. (2016). An $\alpha$ -stable limit theorem under sublinear expectation. Bernoulli 22, 2548–2578.Google Scholar
Biswas, I. H., Chowdhury, I. and Jakobsen, E. R. (2019). On the rate of convergence for monotone numerical schemes for nonlocal Isaacs equations. SIAM J. Numer. Anal. 57, 799–827.CrossRefGoogle Scholar
Biswas, I. H., Jakobsen, E. R. and Karlsen, K. H. (2008). Error estimates for a class of finite difference-quadrature schemes for fully nonlinear degenerate parabolic integro-PDEs. J. Hyperbolic Differ. Equ. 5, 187–219.CrossRefGoogle Scholar
Biswas, I. H., Jakobsen, E. R. and Karlsen, K. H. (2010). Difference-quadrature schemes for non linear degenerate parabolic integro-PDE. SIAM J. Numer. Anal. 48, 1110–1135.CrossRefGoogle Scholar
Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method, Springer, Berlin, 2011.CrossRefGoogle Scholar
Chen, P. and Xu, L. (2019). Approximation to stable law by the Lindeberg principle. J. Math. Anal. Appl. 480, 123338.CrossRefGoogle Scholar
Coclite, G. M., Reichmann, O. and Risebro, N. H. (2016). A convergent difference scheme for a class of partial integro-differential equations modeling pricing under uncertainty. SIAM J. Numer. Anal. 54, 588–605.CrossRefGoogle Scholar
Davydov, Y. and Nagaev, A. V. (2002). On two approaches to approximation of multidimensional stable laws. J. Multivariate Anal. 82, 210–239.CrossRefGoogle Scholar
Fang, X., Peng, S., Shao, Q. and Song, Y. (2019). Limit theorem with rate of convergence under sublinear expectations. Bernoulli 25, 2564–2596.CrossRefGoogle Scholar
Hall, P. (1981). Two-sided bounds on the rate of convergence to a stable law. Prob. Theory Relat. Fields 57, 349–364.Google Scholar
Hall, P. (1981). On the rate of convergence to a stable law. J. Lond. Math. Soc. s2-23, 179–192.CrossRefGoogle Scholar
Häusler, E. and Luschgy, H. (2015). Stable Convergence and Stable Limit Theorems (Prob. Theory Stoch. Modelling 74), Springer, Cham.Google Scholar
Hu, M. and Li, X. (2014). Independence under the G-expectation framework. J. Theoret. Prob. 27, 1011–1020.CrossRefGoogle Scholar
Hu, M. and Peng, S. (2021). G-Lévy processes under sublinear expectations. Prob. Uncert. Quant. Risk 6, 1–22.CrossRefGoogle Scholar
Huang, S. and Liang, G. (2024). A monotone scheme for G-equations with application to the convergence rate of robust central limit theorem. J. Differential Equat. 398, 1–37.CrossRefGoogle Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Jakobsen, E. R., Karlsen, K. H. and La Chioma, C. (2008). Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions. Numer. Math. 110, 221–255.CrossRefGoogle Scholar
Jin, X., Li, X. and Lu, J. (2020). A kernel bound for non-symmetric stable distribution and its applications. J. Math. Anal. Appl. 488, 124063.CrossRefGoogle Scholar
Juozulynas, A. and Paulauskas, V. (1998). Some remarks on the rate of convergence to stable laws. Lith. Math. J. 38, 335–347.CrossRefGoogle Scholar
Krylov, N. V. (1997). On the rate of convergence of finite-difference approximations for Bellman’s equations. St. Petersburg Math. J. 9, 639–650.Google Scholar
Krylov, N. V. (1999). Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies. Electron. J. Prob. 4, 1–19.CrossRefGoogle Scholar
Krylov, N. V. (2000). On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Prob. Theory Relat. Fields 117, 1–16.CrossRefGoogle Scholar
Krylov, N. V. (2020). On Shige Peng’s central limit theorem. Stoch. Process. Appl. 130, 1426–1434.CrossRefGoogle Scholar
Neufeld, A. and Nutz, M. (2017). Nonlinear Lévy processes and their characteristics. Trans. Amer. Math. Soc. 369, 69–95.CrossRefGoogle Scholar
Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality (Camb. Tracts Math. 192). Cambridge University Press.CrossRefGoogle Scholar
Peng, S. (2004). Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. 20, 1–24.CrossRefGoogle Scholar
Peng, S. (2007). G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In Stochastic Analysis and Applications. Abel Symposia, Vol. 2, eds F. E. Benth, G. Di Nunno, T. Lindstrøm, B. Øksendal and T. Zhang. Springer, Berlin, pp. 541–567.CrossRefGoogle Scholar
Peng, S. (2008). Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process. Appl. 118, 2223–2253.CrossRefGoogle Scholar
Peng, S. (2008). A new central limit theorem under sublinear expectations. Preprint, arXiv:0803.2656.Google Scholar
Peng, S. (2019). Law of large numbers and central limit theorem under nonlinear expectations. Prob. Uncertain. Quant. Risk 4, 4.CrossRefGoogle Scholar
Peng, S. (2019). Nonlinear Expectations and Stochastic Calculus under Uncertainty. Springer, New York.CrossRefGoogle Scholar
Song, Y. (2020). Normal approximation by Stein’s method under sublinear expectations. Stoch. Process. Appl. 130, 2838–2850.CrossRefGoogle Scholar
Xu, L. (2019). Approximation of stable law in Wasserstein-1 distance by Stein’s method. Ann. Appl. Prob. 29, 458–504.CrossRefGoogle Scholar