Abstract
In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class \(Lip(\Omega )\). Then we prove generalized strict comparison theorems on the enlarged space \(L_G^1(\Omega )\), which is the Banach completion of \(Lip(\Omega )\) under the G-expectation.
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References
G. Choquet, Theory of capacities, Ann. Inst. Fourier, Grenoble 5 (1953–1954), 131–195 (1955).
M. Crandall, H. Ishii, and P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1–67.
L. Denis, M. Hu, and S. Peng, Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal. 34 (2011), 139–161.
L. Epstein and S. Ji, Ambiguous volatility and asset pricing in continuous time, The Review of Financial Studies 26 (2013), 1740–1786.
M. Hu and S. Peng, On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser. 25 (2009), 539–546.
P. Huber and V. Strassen, Minimax tests and the Neyman-Pearson Lemma for capacity, Ann. Statist. 1 (1973), 252–263.
N.V. Krylov, Nonlinear Parabolic and Elliptic Equations of the Second Order, Translated from the Russian by P. L. Buzytsky [P. L. Buzytski.], Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987.
N.V. Krylov and M.V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1980), 151–164.
S. Peng, Backward SDE and related g–expectations, In: Backward Stochastic Differential Equations, (Paris, 1995–1996), 141–159, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997.
S. Peng, \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô’s type, In: Stochastic Analysis and Applications, 541–567, Abel Symp., 2, Springer, Berlin, 2007.
S. Peng, Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation, Stochastic Process. Appl. 118 (2008), 2223–2253.
S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, arXiv:1002.4546v1.
S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications, In: Proceedings of the International Congress of Mathematicians. Volume I, 393-432, Hindustan Book Agency, New Delhi, 2010.
J. Vorbrink, Financial markets with volatility uncertainty, J. Math. Econom. 53 (2014), 64–78.
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The authors acknowledge fruitful discussions with Professor S. Peng. Xinpeng Li is grateful for financial support from NSF (No. 11601281) and Shandong Provincial Natural Science Foundation (No. ZR2016AQ11). Yiqing Lin is grateful for financial support from the European Research Council (No. 321111) and Shandong Provincial Natural Science Foundation (No. ZR2016FM48).
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Li, X., Lin, Y. Strict comparison theorems under sublinear expectations. Arch. Math. 109, 489–498 (2017). https://doi.org/10.1007/s00013-017-1098-0
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DOI: https://doi.org/10.1007/s00013-017-1098-0