Responsive Distribution of G-normal Random Variables

TL;DR

This paper introduces a novel coupled trinomial tree method to compute $G$-expectations and construct measurement-dependent responsive distributions, validated by convergence proofs and numerical experiments.

Contribution

It proposes a new framework combining backward-forward trinomial trees to approximate $G$-expectations and responsive distributions, with rigorous convergence analysis.

Findings

The coupled scheme converges to the true $G$-expectation.

Numerical results validate the convergence and practical utility of the method.

The approach enables visualization of complex responsive distributions.

Abstract

A $G$-normal random variable $X\sim \mathcal{N}(0,[\underline{\sigma}^2,\overline{\sigma}^2])$ does not admit a unique probability law due to volatility uncertainty. For a given test function $\phi$, the $G$-expectation admits the stochastic control representation$$\mathbb{E}[\phi(X)] = \sup_{\sigma\in[\underline{\sigma},\overline{\sigma}]} {E}\!\left[\phi(X_T^\sigma)\mid X_0^\sigma=0\right] ={E}\!\left[\phi(X_T^\ast)\mid X_0^\ast=0\right].$$ 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 $f_\phi$ such that, for a given test function $\phi$, $$\mathbb{E}[\phi(X)] = \int_{\mathbb{R}} \phi(x)\,f_\phi(x)\,dx.$$ 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.