Stein Type Characterization for $G$-normal Distributions

Abstract

In this article, we provide a Stein type characterization for $G$-normal distributions: Let $\mathcal{N}[\phi ]={\max }_{\mu \in \Theta }\mu [\phi ],\phi \in C_{b,Lip}(\mathbb{R}),$ be a sublinear expectation. $\mathcal{N}$ is $G$-normal if and only if for any $\phi \in C_b^2(\mathbb{R})$, we have \[\int_\mathbb{R}[\frac{x}{2}φ'(x)-G(φ"(x))]μ^φ(dx)=0,\] where ${\mu }^{\phi }$ is a realization of $\phi$ associated with $\mathcal{N}$, i.e., ${\mu }^{\phi }\in \Theta$ and ${\mu }^{\phi }[\phi ]=\mathcal{N}[\phi ]$.

References in corpus (4)

• Some properties on $G$-evaluation and its applications to $G$-martingale decomposition
• A New Central Limit Theorem under Sublinear Expectations
• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
• Dynamically Consistent Nonlinear Evaluations and Expectations