• Bernoulli
  • Volume 22, Number 4 (2016), 2548-2578.

An $\alpha$-stable limit theorem under sublinear expectation

Erhan Bayraktar and Alexander Munk

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For $\alpha\in (1,2)$, we present a generalized central limit theorem for $\alpha$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided a mild but natural additional condition holds. Our approach contrasts with previous arguments for the result in the linear setting which have typically relied upon tools that are non-existent in the sublinear framework, for example, characteristic functions.

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Bernoulli, Volume 22, Number 4 (2016), 2548-2578.

Received: January 2015
Revised: May 2015
First available in Project Euclid: 3 May 2016

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Zentralblatt MATH identifier

generalized central limit theorem partial-integro differential equations stable distribution sublinear expectation


Bayraktar, Erhan; Munk, Alexander. An $\alpha$-stable limit theorem under sublinear expectation. Bernoulli 22 (2016), no. 4, 2548--2578. doi:10.3150/15-BEJ737.

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  • [1] Avellaneda, M., Levy, A. and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
  • [2] Barenblatt, G.I. (1979). Similarity, Self-Similarity, and Intermediate Asymptotics. New York–London: Consultants Bureau [Plenum]. Translated from the Russian by Norman Stein, translation edited by Milton Van Dyke, with a foreword by Ya.B. Zel’dovich [Ja.B. Zel’dovič].
  • [3] Bass, R.F. and Levin, D.A. (2002). Harnack inequalities for jump processes. Potential Anal. 17 375–388.
  • [4] Bayraktar, E. and Munk, A. (2015). Comparing the $G$-normal distribution to its classical counterpart. Commun. Stoch. Anal. 9 1–18.
  • [5] Caffarelli, L. and Silvestre, L. (2009). Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62 597–638.
  • [6] Caffarelli, L. and Silvestre, L. (2011). The Evans–Krylov theorem for nonlocal fully nonlinear equations. Ann. of Math. (2) 174 1163–1187.
  • [7] Caffarelli, L.A. and Cabré, X. (1995). Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications 43. Providence, RI: Amer. Math. Soc.
  • [8] Dolinsky, Y., Nutz, M. and Soner, H.M. (2012). Weak approximation of $G$-expectations. Stochastic Process. Appl. 122 664–675.
  • [9] Hu, M., Ji, S., Peng, S. and Song, Y. (2014). Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion. Stochastic Process. Appl. 124 1170–1195.
  • [10] Hu, M. and Li, X. (2014). Independence under the $G$-expectation framework. J. Theoret. Probab. 27 1011–1020.
  • [11] Hu, M. and Peng, S. (2009). $G$-Lévy processes under sublinear expectations. Preprint. Available at arXiv:0911.3533v1.
  • [12] Hu, Z.-C. and Zhou, L. (2015). Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations. Acta Math. Sin. (Engl. Ser.) 31 305–318.
  • [13] Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff Publishing. With a supplementary chapter by I.A. Ibragimov and V.V. Petrov, translation from the Russian edited by J.F.C. Kingman.
  • [14] Kassmann, M., Rang, M. and Schwab, R.W. (2014). Integro-differential equations with nonlinear directional dependence. Indiana Univ. Math. J. 63 1467–1498.
  • [15] Kassmann, M. and Schwab, R.W. (2014). Regularity results for nonlocal parabolic equations. Riv. Math. Univ. Parma (N.S.) 5 183–212.
  • [16] Kriventsov, D. (2013). $C^{1,\alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels. Comm. Partial Differential Equations 38 2081–2106.
  • [17] Lara, H.C. and Dávila, G. (2012). Regularity for solutions of nonlocal, nonsymmetric equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 833–859.
  • [18] Lara, H.C. and Dávila, G. (2014). Hölder estimates for non-local parabolic equations with critical drift. Preprint. Available at arXiv:1408.0676.
  • [19] Lara, H.C. and Dávila, G. (2014). $C^{\sigma+\alpha}$ estimates for concave, non-local parabolic equations with critical drift. Preprint. Available at arXiv:1408.5149.
  • [20] Li, M. and Shi, Y. (2010). A general central limit theorem under sublinear expectations. Sci. China Math. 53 1989–1994.
  • [21] Luo, P. and Jia, G. (2014). A note on characterizations of $G$-normal distribution. Preprint. Available at arXiv:1402.4631.
  • [22] Neufeld, A. and Nutz, M. (2015). Nonlinear Lévy processes and their characteristics. Trans. Amer. Math. Soc. To appear.
  • [23] Osuka, E. (2013). Girsanov’s formula for $G$-Brownian motion. Stochastic Process. Appl. 123 1301–1318.
  • [24] Paczka, K. (2012). Itô calculus and jump diffusions for $G$-Lévy processes. Preprint. Available at arXiv:1211.2973.
  • [25] Paczka, K. (2014). $G$-martingale representation in the $G$-Lévy setting. Preprint. Available at arXiv:1404.2121.
  • [26] Peng, S. (2007). Law of large numbers and central limit theorem under nonlinear expectations. Preprint. Available at arXiv:math/0702358.
  • [27] Peng, S. (2007). $G$-Brownian motion and dynamic risk measure under volatility uncertainty. Preprint. Available at arXiv:0711.2834.
  • [28] Peng, S. (2008). A new central limit theorem under sublinear expectations. Preprint. Available at arXiv:0803.2656.
  • [29] Peng, S. (2009). Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A 52 1391–1411.
  • [30] Peng, S. (2010). Nonlinear expectations and stochastic calculus under uncertainty. Preprint. Available at arXiv:1002.4546.
  • [31] Peng, S., Song, Y. and Zhang, J. (2014). A complete representation theorem for $G$-martingales. Stochastics 86 609–631.
  • [32] Ren, L. (2013). On representation theorem of sublinear expectation related to $G$-Lévy process and paths of $G$-Lévy process. Statist. Probab. Lett. 83 1301–1310.
  • [33] Schwab, R.W. and Silvestre, L. (2014). Regularity for parabolic integro-differential equations with very irregular kernels. Preprint. Available at arXiv:1412.3790.
  • [34] Serra, J. (2014). $C^{\sigma+\alpha}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels. Preprint. Available at arXiv:1405.0930.
  • [35] Silvestre, L. (2011). On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion. Adv. Math. 226 2020–2039.
  • [36] Silvestre, L. (2014). Regularity estimates for parabolic integro-differential equations and applications. In Proceedings of the International Congress of Mathematicians (Seoul, 2014) (S.Y. Jang, Y.R. Kim, D.-W. Lee andI. Yie, eds.) 873–894. Seoul: Kyung Moon SA. Available at
  • [37] Soner, H.M., Touzi, N. and Zhang, J. (2011). Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 265–287.
  • [38] Song, Y. (2011). Some properties on $G$-evaluation and its applications to $G$-martingale decomposition. Sci. China Math. 54 287–300.
  • [39] Xu, J., Shang, H. and Zhang, B. (2011). A Girsanov type theorem under $G$-framework. Stoch. Anal. Appl. 29 386–406.
  • [40] Zhang, D. and Chen, Z. (2014). A weighted central limit theorem under sublinear expectations. Comm. Statist. Theory Methods 43 566–577.
  • [41] Nonlocal Equations Wiki. Available at Accessed 2015-01-25.