Filtration-consistent nonlinear expectations and related \(g\)-expectations
type: Item
instance of: scholarly article
title: Filtration-consistent nonlinear expectations and related \(g\)-expectations
zbMATH Open document ID: 1007.60057
author: François Coquet
author: Ying Hu
author: Jean Mémin
author: Shige Peng
publication date: 2002-12-01T00:00:00Z
review text: Given a probability space \((\Omega,{\mathcal F},P)\) endowed with a \(d\)-dimensional Brownian motion \(W\) and an application \(g\) from \((\Omega\times [0,T]\times \mathbb{R}\times \mathbb{R}^d)\) to \(\mathbb{R}\) satisfying convenient assumptions, one can define a \(g\)-expectation (resp. -conditional expectation) by \[ \forall X \in L^2(\Omega, {\mathcal F},P),\;\xi_g(X)= Y_0\text{ (resp. }\xi_g (X\mid{\mathcal F}_t) =Y_t), \] where \((Y,Z)\) is the solution of the backward stochastic differential equation \[ Y_t=X+ \int^T_tg (s,Y_s,Z_s)dx- \int^T_tZ_sdW_s,\quad t\in [0,T], \] and \(({\mathcal F}_t)\) is the filtration generated by \(W\) [see \textit{S. Peng}, in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 141-159 (1997; Zbl 0892.60066)]. In this paper, the authors introduce the notion of nonlinear expectation as a map \(\xi\) from \(L^2(\Omega, {\mathcal F}, P)\) to \(\mathbb{R}\) that satisfies (1) \(X_1\leq X_2\) a.s. \(\Rightarrow\xi (X_1)\leq \xi (X_2)\), (2) if \(X_1\leq X_2\) a.s., then \(\xi(X_1)= \xi(X_2) \Leftrightarrow X_1= X_2\) a.s., (3) \(\xi(c)=c\), for all \(c\) constant. The notions of conditional nonlinear expectation and nonlinear martingale are derived. Various properties are studied. The main result of the paper is that, under some rather general assumptions, every nonlinear martingale is a \(g\)-martingale.
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1837121
zbMATH Keywords: backward stochastic differential equation
zbMATH Keywords: nonlinear expectation
zbMATH Keywords: \(g\)-martingale
zbMATH Keywords: nonlinear martingale Doob-Meyer decomposition
reviewed by: Catherine Rainer
MaRDI profile type: Publication
full work available at URL: s004400100172
OpenAlex ID: W2052855777
DOI: 10.1007/S004400100172
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1837121