A result on the probability measures dominated by \(g\)-expectation
Given a \(d\)-dimensional Brownian motion \(W\), consider the backward stochastic differential equation \[ dY_{t}^{\xi}=-g(t,Y_{t}^{\xi} ,Z_{t}^{\xi})dt+Z_{t}^{\xi}dW_{t}, \;t\in[0,T], \qquad Y_{T}^{\xi}=\xi\in L^{2}(\Omega,\mathcal{F}_{T}^{W},P), \] where \(g:[0,T]\times\Omega\times R\times R^{d}\to R\) is a coefficient satisfying in addition to standard assumptions the relation \(g(t,y,0)=0\) for all \(t\in[0,T]\). For such \(g\), motivated by the study of expected utility in finance, \textit{S. Peng} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 141--159 (1997; Zbl 0892.60066)] has introduced the notion of \(g\)-expectation of a random variable \(\xi\in L^{2}(\Omega,\mathcal{F}_{T}^{W},P)\) as \(\mathcal{E}_{g}(\xi):=Y_{0}^{\xi}\). Since his pioneering work several authors have investigated the properties of \(g\)-investigations. The authors of the present paper study a problem which arouses naturally in this context [cf., for instance; \textit{Z. Chen} and \textit{L. G. Epstein} [Econometrica 70, 1403--1443 (2002)]. The problem consists to compare the sets \[ S_{1}=\{Q~| ~E_{Q} [\xi]\leq\mathcal{E}_{g}(\xi),\;\forall\xi\in L^{2}(\Omega,\mathcal{F} _{T}^{W},P)\} \] and \[ S_{2}= \left\{Q^{\theta}~\biggl| ~Q^{\theta}=\exp\left(\int_0^T \theta_s dW_s -\frac{1}{2}\int_0^T| \theta_{s}| ^{2}ds\right), \;\theta\;\mathbb{F}^{W}\text{-adapted and bounded by }\mu \right\}, \] where \(\mu\) is a given positive real and \(g(z)=\mu\cdot| z| \), \(z\in R^{d}\). The authors show that the both sets coincide.
backward stochastic differential equation
scientific article; zbMATH DE number 2113814
scientific article