Under the basic assumptions of $g$-expectations defined in Chen and Wang (2000), we establish the one-to-one correspondence between generators of backward stochastic differential equations (BSDEs for short) and the convexity (resp. conditional convexity, ${\mathcal{F}}_t$-convexity) of $g$-expectations, respectively. We also obtain the relationship between generators of BSDEs and the sublinearity (resp. conditional sublinearity, ${\mathcal{F}}_t$-sublinearity) of $g$-expectations, respectively. Moreover, we provide the reasonable assumptions on generators of BSDEs for the further study on dynamic risk measures by applying the theory of $g$-expectations.

中文翻译：

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g - 期望的凸性和次线性

在基本假设下 $G$-在 Chen 和 Wang (2000) 中定义的期望，我们建立了后向随机微分方程（简称 BSDE）和凸性（分别是条件凸性， ${\mathcal{F}}_{吨}$-凸性）的 $G$-期望，分别。我们还获得了 BSDE 的生成器与次线性（分别是条件次线性， ${\mathcal{F}}_{吨}$-次线性）的 $G$-期望，分别。此外，我们通过应用理论，为进一步研究动态风险度量提供了关于 BSDE 生成器的合理假设。 $G$-期望。