Nonlinear Expectations and Stochastic Calculus under Uncertainty

Abstract

In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls.

149 pages

Cited by in corpus (14)

• Constructing Sublinear Expectations on Path Space
• Loss of regularity for Kolmogorov equations
• Random G-expectations
• A monotone scheme for high-dimensional fully nonlinear PDEs
• On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
• The convergence of the sums of independent random variables under the sub-linear expectations
• Local time and Tanaka formula for G-Brownian Motion
• Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation
• Properties of solutions of stochastic differential equations driven by the G-Brownian motion
• Supermartingale Decomposition Theorem under G-expectation
• Path Independence of Additive Functionals for SDEs under G-framework
• Lévy's martingale characterization and reflection principle of $G$-Brownian motion
• Characterizations of processes with stationary and independent increments under $G$-expectation
• Some inequalities and limit theorems under sublinear expectations