Comparison, Uniqueness and Continuity on Energy Sublevel Sets 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, 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 $\sigma$.

1. (Comparison)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, let $u$ be an envelope viscosity subsolution and $v$ an envelope viscosity supersolution of $(\mathrm{E})$, each with some shift range. Then $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.

2. (Uniqueness)¶ Any two bounded envelope viscosity solutions $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ of $(\mathrm{E})$, each with some shift range, satisfy $u(\kappa_{d}(\mu))=v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.

3. (Continuity on energy sublevel sets)¶ Let $u:\Sigma^{2}_{d}\to\mathbb{R}$ be a bounded envelope viscosity solution of $(\mathrm{E})$ with some shift range, and let $c$ be real. Then the function $\mu\mapsto u(\kappa_{d}(\mu))$ on the set $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ is continuous for the metric $W_{2}$.