Filtration-consistent nonlinear expectations and related \(g\)-expectations
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.
backward stochastic differential equation
nonlinear martingale Doob-Meyer decomposition
scientific article; zbMATH DE number 1837121