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Nonlinear Expectations and Stochastic Calculus under Uncertainty

Shige Peng
Publication Date: 
Number of Pages: 
Probability Theory and Stochastic Modelling
[Reviewed by
Nikos Halidias
, on
This book is an extension of the author's Lecture notes. The author here taken the notion of nonlinear expectation as a fundamental notion of an axiomatic system. As a result,  he is able to develop many new and fundamental results such as nonlinear law of large numbers, central limit theorem, the theory of Brownian motion under nonlinear expectation, and the corresponding new stochastic analysis of Ito's type. 
The author presents the basic notion of sublinear expectations and the corresponding sublinear expectation spaces. He gives the representation theorem of a sublinear expectation and the notions of distributions and independence within the framework of sublinear expectations. As a fundamentally important example, he introduces the notion of coherent risk measures in finance.  He introduces two types of distributions, namely, maximal distribution and a new type of nonlinear normal distribution, \( G \) - normal distribution in the theory of sublinear expectations.  He then presents the law of large numbers (LLN) and central limit theorem (CLT) under sublinear expectations.
The author introduces the concept of \( G \)-Brownian motion, studies its properties, and constructs Ito's integral with respect to \( G \)-Brownian motion. A very interesting feature of the \( G \)-Brownian motion is that its quadratic process also has independent increments which are identically distributed.  The corresponding \( G \)-Ito's formula is presented.  The author also considers stochastic differential equations and backward stochastic differential equations driven by \( G \)-Brownian motion.


Nikos Halidias is a professor of mathematics at the University of the Aegean, Department of Statistics and Actuarial - Financial Mathematics, Greece. His research and teaching interests are in Differential Equations and Stochastic Analysis.