Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall

In the setting of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation, suppose $\mathcal{D}$ is nonempty, for bounded $w:\Sigma^{2}_{d}\to\mathbb{R}$ let $w^{\mp}_{\delta}$ be its penalty envelopes as in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §envelopes, and assume that the Hamiltonian $\mathcal{H}$ satisfies the structure condition at bounded positions, is uniformly continuous in the momentum at bounded positions, and absorbs shifts at noise level $\sigma$. Let $b\ge0$ and $\delta_{0}>0$ be real.

¶There is a function $\epsilon:(0,\delta_{0}]\to[0,\infty)$ with $\epsilon(\delta)\to0$ as $\delta\to0$ (that is, for every real $\eta>0$ there is a real $\delta_{1}>0$ with $\epsilon(\delta)<\eta$ whenever $\delta$ lies in the domain of $\epsilon$ and $\delta<\delta_{1}$) such that the following holds. For every real $\delta$ with $0<\delta\le\delta_{0}$, every envelope viscosity subsolution $u$ and every envelope viscosity supersolution $v$ of $(\mathrm{E})$ with shift range $\delta_{0}$ and $|u|\le b$, $|v|\le b$ on $\Sigma^{2}_{d}$,

$$u^{-}_{\delta}(\mu)\le v^{+}_{\delta}(\mu)+\epsilon(\delta)\qquad\text{for every }\mu\in\mathcal{D}.$$