Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations

This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Itô’s type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.

[1]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[2]  Larry G. Epstein,et al.  Ambiguity, risk, and asset returns in continuous time , 2000 .

[3]  Bo Zhang,et al.  Martingale characterization of G-Brownian motion , 2009 .

[4]  Shige Peng,et al.  NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS , 2005 .

[5]  Shige Peng,et al.  A New Central Limit Theorem under Sublinear Expectations , 2008, 0803.2656.

[6]  Shige Peng,et al.  Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims , 2004 .

[7]  Lihe Wang.  On the regularity theory of fully nonlinear parabolic equations: II , 1992 .

[8]  S. Peng,et al.  Filtration-consistent nonlinear expectations and related g-expectations , 2002 .

[9]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[10]  S. Peng.  Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation , 2006, math/0601699.

[11]  George Yin,et al.  Asymptotic Expansions of Transition Densities for Hybrid Jump-diffusions , 2004 .

[12]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[13]  Klemens Szaniawski,et al.  Formal methods in the methodology of empirical sciences : proceedings of the Conference for Formal Methods in the Methodology of Empirical Sciences, Warsaw, June 17-21, 1974 , 1976 .

[14]  L. Denis,et al.  A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN THE PRESENCE OF MODEL UNCERTAINTY , 2006, math/0607111.

[15]  S. Peng.  G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type , 2006, math/0601035.

[16]  Mingshang Hu,et al.  Explicit solutions of G-heat equation with a class of initial conditions by G-Brownian motion , 2009, 0907.2748.

[17]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[18]  N. Karoui,et al.  Backward Stochastic Differential Equations , 1997 .

[19]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[20]  Shige Peng,et al.  Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths , 2008, 0802.1240.

[21]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[22]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[23]  S. Peng.  Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyers type , 1999 .

[24]  S. Peng.  G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty , 2007, 0711.2834.

[25]  Emanuela Rosazza Gianin,et al.  Risk measures via g-expectations , 2006 .

[26]  H. Soner,et al.  Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs , 2005, math/0509295.

[27]  N. El Karoui,et al.  Pricing, Hedging and Optimally Designing Derivatives via Minimization of Risk Measures , 2007, 0708.0948.

[28]  Shige Peng,et al.  On representation theorem of G-expectations and paths of G-Brownian motion , 2009, 0904.4519.

[29]  P. Levy,et al.  Calcul des Probabilites , 1926, The Mathematical Gazette.