Proof of Comparison, Uniqueness and Continuity on Energy Sublevel Sets for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall

Each result cited below is universally quantified over the data in its own statement.

Step 0 (Sandwich). Let $w:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded and $\delta>0$ real. The function $w\circ\kappa_{d}-\delta\mathcal{E}$ on $\mathcal{D}$ is bounded above near each point by The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §upper, so Properties of the Upper Semicontinuous Envelope §bounds, for the nonempty set $\mathcal{D}$ in the metric space $(\Sigma_{d,R},W_{2})$ of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, gives $w(\kappa_{d}(\mu))-\delta\,\mathcal{E}(\mu)\le w^{-}_{\delta}(\mu)$ for every $\mu\in\mathcal{D}$; likewise The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §lower and Properties of the Lower Semicontinuous Envelope, by Duality §bounds give $w^{+}_{\delta}(\mu)\le w(\kappa_{d}(\mu))+\delta\,\mathcal{E}(\mu)$. Moreover $w^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$ by Properties of the Upper Semicontinuous Envelope §usc, and $w^{+}_{\delta}$ is lower semicontinuous on $\mathcal{D}$ by Properties of the Lower Semicontinuous Envelope, by Duality §lsc.

Step 1 (Common data). Let $u$ be a bounded envelope subsolution with shift range $\delta_{u}$ and $v$ a bounded envelope supersolution with shift range $\delta_{v}$, and put $\delta_{0}=\min\{\delta_{u},\delta_{v}\}$ and $b=\max\{b_{u},b_{v}\}$ with $|u|\le b_{u}$ and $|v|\le b_{v}$. Every $\delta$ with $0<\delta\le\delta_{0}$ satisfies $\delta\le\delta_{u}$ and $\delta\le\delta_{v}$, so the conditions of Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub and Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super for all such $\delta$ are among those assumed: $u$ is an envelope subsolution and $v$ an envelope supersolution with shift range $\delta_{0}$, and $|u|\le b$, $|v|\le b$. The hypotheses of Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §fixed-level on the setting, on $\mathcal{D}$ and on $\mathcal{H}$ are those of the present theorem; $b\ge0$ because a bound of a bounded function is nonnegative, and $\delta_{0}>0$ as the least of two positive reals. Let $\epsilon:(0,\delta_{0}]\to[0,\infty)$ be the function of that lemma for these $b$ and $\delta_{0}$.

Claim 1 (Comparison). Fix $\mu\in\mathcal{D}$. For every $\delta\in(0,\delta_{0}]$, Step 0 and Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §fixed-level give

$$u(\kappa_{d}(\mu))-\delta\,\mathcal{E}(\mu)\le u^{-}_{\delta}(\mu)\le v^{+}_{\delta}(\mu)+\epsilon(\delta)\le v(\kappa_{d}(\mu))+\delta\,\mathcal{E}(\mu)+\epsilon(\delta),$$

that is, $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))+2\delta\,\mathcal{E}(\mu)+\epsilon(\delta)$. Let $\eta>0$ be real. Since $\epsilon(\delta)\to0$ as $\delta\to0$ in the sense of Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §fixed-level, there is a real $\delta_{1}>0$ with $\epsilon(\delta)<\eta/2$ for every $\delta$ in $(0,\delta_{0}]$ with $\delta<\delta_{1}$. Choose a real $\delta$ with $0<\delta\le\delta_{0}$, $\delta<\delta_{1}$ and $2\delta\,|\mathcal{E}(\mu)|<\eta/2$ (for instance half the least of $\delta_{0}$, $\delta_{1}$ and $\eta/(4|\mathcal{E}(\mu)|+4)$). Then $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))+\eta$. As $\eta>0$ was arbitrary, $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above.

Claim 2 (Uniqueness). Each of $u,v$ is by Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §solution both an envelope subsolution and an envelope supersolution with its shift range. Claim 1 applied to the pair $(u,v)$ and to the pair $(v,u)$ gives $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$ and $v(\kappa_{d}(\mu))\le u(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$, hence equality by antisymmetry of the order.

Claim 3 (Continuity on energy sublevel sets). Let $u$ be a bounded envelope solution with shift range $\delta_{0}$ and $|u|\le b$, let $c$ be real and $S_{c}=\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$; if $S_{c}$ is empty there is nothing to prove. By Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §solution, $u$ is both an envelope subsolution and an envelope supersolution with shift range $\delta_{0}$, so, as in Step 1, Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §fixed-level applies with $v=u$ for these $b$ and $\delta_{0}$ and gives a function $\epsilon$ on $(0,\delta_{0}]$ tending to $0$ with

$$u^{-}_{\delta}(\lambda)\le u^{+}_{\delta}(\lambda)+\epsilon(\delta)\qquad\text{for every }\lambda\in\mathcal{D}\text{ and }\delta\in(0,\delta_{0}].\tag{C}$$

Fix $\mu\in S_{c}$ and a real $\eta>0$. Let $\delta_{1}>0$ be such that $\epsilon(\delta)<\eta/3$ for every $\delta\in(0,\delta_{0}]$ with $\delta<\delta_{1}$, and let $\delta$ be half the least of $\delta_{0}$, $\delta_{1}$ and $\eta/(3|c|+3|\mathcal{E}(\mu)|+3)$; then $\delta\in(0,\delta_{0}]$, $\epsilon(\delta)<\eta/3$ and $\delta\,(|c|+|\mathcal{E}(\mu)|)<\eta/3$. By Step 0 and the definition of upper semicontinuity (Upper Semicontinuous Function on a Subset of a Metric Space) there is a real $r_{1}>0$ with $u^{-}_{\delta}(\nu)<u^{-}_{\delta}(\mu)+\eta/3$ for every $\nu\in\mathcal{D}$ with $W_{2}(\mu,\nu)<r_{1}$, and by Lower Semicontinuous Function on a Subset of a Metric Space there is a real $r_{2}>0$ with $u^{+}_{\delta}(\mu)-\eta/3<u^{+}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$ with $W_{2}(\mu,\nu)<r_{2}$. Let $\nu\in S_{c}$ with $W_{2}(\mu,\nu)<\min\{r_{1},r_{2}\}$. Since $\mathcal{E}(\nu)\le c\le|c|$ and $\delta>0$, Step 0 and (C) at $\mu$ give

$$u(\kappa_{d}(\nu))\le u^{-}_{\delta}(\nu)+\delta\,\mathcal{E}(\nu)<u^{-}_{\delta}(\mu)+\tfrac{\eta}{3}+\delta|c|\le u^{+}_{\delta}(\mu)+\epsilon(\delta)+\tfrac{\eta}{3}+\delta|c|\le u(\kappa_{d}(\mu))+\delta\,\mathcal{E}(\mu)+\epsilon(\delta)+\tfrac{\eta}{3}+\delta|c|<u(\kappa_{d}(\mu))+\eta,$$

and, since $-\delta\,\mathcal{E}(\nu)\ge-\delta|c|$,

$$u(\kappa_{d}(\nu))\ge u^{+}_{\delta}(\nu)-\delta\,\mathcal{E}(\nu)>u^{+}_{\delta}(\mu)-\tfrac{\eta}{3}-\delta|c|\ge u^{-}_{\delta}(\mu)-\epsilon(\delta)-\tfrac{\eta}{3}-\delta|c|\ge u(\kappa_{d}(\mu))-\delta\,\mathcal{E}(\mu)-\epsilon(\delta)-\tfrac{\eta}{3}-\delta|c|>u(\kappa_{d}(\mu))-\eta .$$

Hence $|u(\kappa_{d}(\nu))-u(\kappa_{d}(\mu))|<\eta$ for every $\nu\in S_{c}$ with $W_{2}(\mu,\nu)<\min\{r_{1},r_{2}\}$. As $\mu\in S_{c}$ and $\eta>0$ were arbitrary, the function $\mu\mapsto u(\kappa_{d}(\mu))$ on $S_{c}$, with the restriction of $W_{2}$ and the metric of $\mathbb{R}$ fixed in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, is continuous.