Envelope Viscosity Subsolutions, Supersolutions and 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, let $u:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, with penalty envelopes $u^{\mp}_{\delta}$ as in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §envelopes, and with the shifted plans $\pi\oplus t\,\Xi(\mu)$ and the norm $\lVert\Xi(\mu)\rVert_{2}$ of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §shifts, and let $\delta_{0}>0$ be real, the shift range. When the Hamiltonian $\mathcal{H}$ is taken to be a specific function $\Sigma^{2}_{2d}\to\mathbb{R}$, the notions below refer to the equation $(\mathrm{E})$ of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data with that Hamiltonian.

1. (Subsolution)¶ $u$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$ if the following holds for every real $\delta$ with $0<\delta\le\delta_{0}$, every function $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, every $\mu\in\mathcal{D}$ such that

$$u^{-}_{\delta}(\nu)-\varphi\bigl(\kappa_{d}(\nu)\bigr)<u^{-}_{\delta}(\mu)-\varphi\bigl(\kappa_{d}(\mu)\bigr)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }\nu\ne\mu,$$

and every bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$: one has $\mu\in\mathcal{D}_{\Xi}$, and for this $\mu$

$$\rho\bigl(u^{-}_{\delta}(\mu)+\delta\,\mathcal{E}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)+\delta\,\lVert\Xi(\mu)\rVert_{2}^{2}\Bigr)\le0.$$

2. (Supersolution)¶ $u$ is an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$ if the following holds for every real $\delta$ with $0<\delta\le\delta_{0}$, every function $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, every $\mu\in\mathcal{D}$ such that

$$u^{+}_{\delta}(\nu)-\varphi\bigl(\kappa_{d}(\nu)\bigr)>u^{+}_{\delta}(\mu)-\varphi\bigl(\kappa_{d}(\mu)\bigr)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }\nu\ne\mu,$$

and every bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{-}\varphi(\kappa_{d}(\mu))$: one has $\mu\in\mathcal{D}_{\Xi}$, and for this $\mu$

$$\rho\bigl(u^{+}_{\delta}(\mu)-\delta\,\mathcal{E}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)-\delta\,\lVert\Xi(\mu)\rVert_{2}^{2}\Bigr)\ge0.$$

3. (Solution)¶ $u$ is an envelope viscosity solution of $(\mathrm{E})$ with shift range $\delta_{0}$ if it is both an envelope viscosity subsolution and an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$.