Dynamically consistent nonlinear evaluations with their generating functions in \(L^p\)
type: Item
instance of: scholarly article
title: Dynamically consistent nonlinear evaluations with their generating functions in \(L^p\)
zbMATH Open document ID: 1271.60068
author: Feng Hu
publication date: 2013-03-28T00:00:00Z
review text: The paper is concerned with an important problem of stochastic finance: the evaluation of the discounted value of a derivative asset. This problem is treated as a dynamically consistent nonlinear evaluation (expectation) and the first results in this direction was given by \textit{S. Peng} [C. R., Math., Acad. Sci. Paris 339, No. 8, 585--589 (2004; Zbl 1065.60087); Acta Math. Appl. Sin., Engl. Ser. 20, No. 2, 191--214 (2004; Zbl 1061.60063)] in the form the well-known \(g\)-evaluation. The author follows a result of \textit{F. Hu} and \textit{Z. Chen} [Stat. Probab. Lett. 80, No. 3--4, 191--195 (2010; Zbl 1181.60084)] and study dynamically consistent nonlinear evaluations in \({\mathcal L}\); they obtain that, under a domination condition, an \({\mathcal F}_t\)-consistent evaluation is an \({\mathcal E}^g\)-evaluation in \({\mathcal L}\), which generalizes the famous results given by S. Peng [loc. cit.] in 2004--2005. Furthermore, without the assumption that the generating function \(g(t,\omega,y,z)\) is continuous with respect to \(t\), the author gives some useful characterizations of \({\mathcal E}^g\)-evaluations and extends some similar results given by \textit{E. Rosazza Gianin} [Insur. Math. Econ. 39, No. 1, 19--34 (2006; Zbl 1147.91346)] and \textit{L. Jiang} [Ann. Appl. Probab. 18, No. 1, 245--258 (2008; Zbl 1145.60032)]. In the second section, the author shows that, for each well-defined function \(g\), the solution of a corresponding backward stochastic differential equation satisfies some adequate conditions of dynamically consistent nonlinear evaluations. An important result is given in Theorem 2.3 which extends similar results given by F. Hu and Z. Chen [loc. cit.]. In the next section, the author proves (Theorem 3.1) the first main result which we mentioned above. In Section 4, some important results (Propositions 4.1--4.6) are proved which show that the behavior of a \(g\)-evaluation is perfectly reflected by its generating function \(g\). Also, two …
reviewed by: Romeo Negrea
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 93E03
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 6149098
zbMATH Keywords: backward stochastic differential equations
zbMATH Keywords: dynamically consistent nonlinear evaluation
zbMATH Keywords: \(g\)-expectation
zbMATH Keywords: dynamic risk measure
zbMATH Keywords: domination condition
zbMATH Keywords: generating functions
zbMATH Keywords: derivatives pricing
zbMATH Keywords: stochastic finance
MaRDI profile type: Publication
full work available at URL: s10114-013-1715-1
OpenAlex ID: W2163955104
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DOI: 10.1007/S10114-013-1715-1
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6149098