A type of filtration-consistent nonlinear expectations under a Brownian filtration was introduced under the name “\(g\)-expectation” by S. Peng [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 141-159 (1997; Zbl 0892.60066)]. These \(g\)-expectations on a probability space \((\Omega,\mathcal F,P)\) are maps \(\mathcal E:L^2(\Omega,\mathcal F, P)\mapsto\mathbb R\) and can be considered as nonlinear extensions of the well-known Girsanov transformations. It is a nonlinear mapping, but it preserves almost all other properties of the classical linear expectations. In this paper, from a general definition of nonlinear expectations, viewed as operators preserving monotonicity and constants, under rather general assumptions, the authors derive the notions of conditional nonlinear expectation and nonlinear martingales, and prove that any such nonlinear martingale can be represented as the solution of a backward stochastic differential equation, and in particular admits continuous paths. In other words, it is a \(g\)-martingale. The main objective of the paper is an answer to the following question: Is the notion of \(g\)-expectation general enough to represent all “enough regular” filtration-consistent nonlinear expectations? The answer gives the following theorem: If for a large enough \(\mu> 0\) a nonlinear expectation \(\mathcal E[\cdot]\) is dominated by the ‘\(\mu|z|\)-expe ctation’ \(\mathcal E^\mu[\cdot]\) (that is, the \(g\)-expectation defined by \(g(z) =\mu|z|\)), and if \(\mathcal E[X + \eta\mid \mathcal F_t]=\mathcal E[X\mid \mathcal F_t]+\eta\) for all \(\mathcal F_t\)-measurable \(\eta\) (i.e., the nonlinearity of \(\mathcal E[\cdot]\) is only to the risk), then there exists a unique \(g\) such that \(\mathcal E[\cdot]\) is the nonlinear expectation defined by \(g\), still according to the definition of S. Peng. The main tool is the decomposition theorem for \(g\)-supermartingales, proved by S. Peng, developed here along a new version suitable for continuous \(\mathcal E\)-supermartingales.
For the entire collection see [Zbl 0988.00099].