Perron's Method for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall: Existence and Well-Posedness

In the setting of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation, let $\rho,\sigma,R>0$ be the discount rate, the noise intensity and the wall radius; let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, be the wall-confined free energy with radius $R$ of the free entropy penalty $(\mathcal{D}_{0},\mathcal{E}_{0})$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$; let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ be the Hamiltonian; and let $(\mathrm{E})$ be the discounted Hamilton--Jacobi--Bellman equation with free Langevin noise in a wall, in score form, with these data, all as fixed in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data. Suppose $\mathcal{D}$ is nonempty, assume that the Hamiltonian $\mathcal{H}$ absorbs shifts at noise level $\sigma$, is shift-semicontinuous at noise level $\sigma$ and is bounded at zero momentum at bounded positions, and assume that $\mathcal{E}$ has closed score and has regular plan subgradients. Subsolutions, supersolutions and solutions are envelope viscosity ones.

1. (Perron's method)¶ Let $g_{-},g_{+}:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, let $g_{-}$ be a subsolution and $g_{+}$ a supersolution of $(\mathrm{E})$, each with some shift range, and suppose $g_{-}\le g_{+}$ on $\Sigma^{2}_{d}$. Then there is a bounded solution $u:\Sigma^{2}_{d}\to\mathbb{R}$ of $(\mathrm{E})$ with some shift range and with $g_{-}\le u\le g_{+}$ on $\Sigma^{2}_{d}$.

2. (Existence)¶ $(\mathrm{E})$ has a bounded solution $u:\Sigma^{2}_{d}\to\mathbb{R}$ with some shift range.

Suppose moreover that $\mathcal{H}$ satisfies the structure condition at bounded positions and is uniformly continuous in the momentum at bounded positions.

3. (Uniqueness)¶ Any two bounded 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 $\mu\in\mathcal{D}$.

4. (Continuity)¶ For every bounded solution $u$ of $(\mathrm{E})$ with some shift range and every real $c$, 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 $W_{2}$ of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric.