A $G$-normal random variable $X\sim \mathcal{N}(0,[\underlineσ^2,\overlineσ^2])$ does not admit a unique probability law due to volatility uncertainty. For a given test function $φ$, the $G$-expectation admits the stochastic control representation$$\mathbb{E}[φ(X)] = \sup_{σ\in[\underlineσ,\overlineσ]} {E}\!\left[φ(X_T^σ)\mid X_0^σ=0\right] ={E}\!\left[φ(X_T^\ast)\mid X_0^\ast=0\right].$$ This formulation interprets the nonlinear expectation as a linear expectation under the law induced by the optimally controlled diffusion $X^\ast$, namely, the terminal law of $X_T^\ast$. This observation motivates the notion of a \emph{responsive distribution}, a measurement-dependent probability density $f_φ$ such that, for a given test function $φ$, $$\mathbb{E}[φ(X)] = \int_{\mathbb{R}} φ(x)\,f_φ(x)\,dx.$$ Based on this viewpoint, we propose a coupled backward--forward trinomial tree framework for computing the $G$-expectation and constructing the corresponding responsive distribution. The backward trinomial tree discretizes the associated stochastic optimal control problem and yields approximations of the value function (i.e., the $G$-expectation) and the optimal feedback control, while the forward trinomial tree propagates the induced transition probabilities and produces a discrete approximation of the responsive distribution. We establish rigorous convergence results for both components of the method. Numerical results not only validate the theoretical convergence of the coupled schemes but also provide a powerful, practical sampling tool to visualize the complex responsive distributions under various measurements.

翻译：由于波动不确定性，$G$-正态随机变量$X\sim \mathcal{N}(0,[\underlineσ^2,\overlineσ^2])$并不具有唯一的概率律。对于给定的测试函数$φ$，$G$-期望满足随机控制表示：$$\mathbb{E}[φ(X)] = \sup_{σ\in[\underlineσ,\overlineσ]} {E}\!\left[φ(X_T^σ)\mid X_0^σ=0\right] ={E}\!\left[φ(X_T^\ast)\mid X_0^\ast=0\right].$$ 该公式将非线性期望解释为在由最优控制扩散过程$X^\ast$诱导的律（即$X_T^\ast$的终期律）下的线性期望。这一观察启发了\textit{响应分布}的概念：一种依赖于测量的概率密度函数$f_φ$，使得对于给定测试函数$φ$，有$$\mathbb{E}[φ(X)] = \int_{\mathbb{R}} φ(x)\,f_φ(x)\,dx.$$ 基于此观点，我们提出了一种耦合的后向-前向三项树框架，用于计算$G$-期望并构建相应的响应分布。后向三项树离散化相关的随机最优控制问题，得到值函数（即$G$-期望）和最优反馈控制的逼近；而前向三项树传播诱导的转移概率，并生成响应分布的离散逼近。我们为该方法的两部分均建立了严格的收敛性结果。数值结果不仅验证了耦合格式的理论收敛性，还提供了一种强大实用的采样工具，用于可视化各种测量下复杂的响应分布。