Chaotic expansion in the \(G\)-expectation space (Q2864641)

In 2010, S.~Peng introduced a new notion of a nonlinear expectation space. He was motivated by some uncertainty problems in finance. This new expectation, called \(G\)-expectation, can take the uncertainty into consideration in a systematic way. The \(G\)-expectation attracted many authors to consider new problems, among others, the \(G\)-normal random variable under the framework of a \(G\)-expectation space.NEWLINENEWLINEThe paper is an interesting contribution of the theory of the \(G\)-expectation space. The note consists in seven sections and references. The first section has introductory character, where the aim of the paper is described. The second section presents notations and preliminaries of the theory of \(G\)-expectation and \(G\)-Brownian motion. In the third section, the \(G\)-stochastic integral and its properties are recalled. Section 4 contains the main results of the paper. Theorem 4.1 gives the equivalence of the orthogonality of Wiener chaos and Theorem 4.3 supplies the result of \(G\)-Wiener chaos. Section 5 provides some relationship between Hermite polynomials and \(G\)-stochastic multiple integrals.NEWLINENEWLINEThe proofs of Theorems 4.1 and 4.3 are given in Sections 6 and 7, respectively. The proofs are clearly written with details. The paper is finished with 13 items representative for the subject under consideration.