Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces and the test functions $F\in C^{1}(E_{d})$, the maps $F\circ\iota_{d}$ and the gradients $DF$ of Unitary Laws with Free Unitary Noise: Standing Data §test-functions, let $H:\mathcal{L}_{d}\times E_{d}\to\mathbb{R}$ be a map, the Hamiltonian, let $\rho,\beta$ be the constants, $f$ the source and $\widehat{\Theta}\lambda$ the noise direction, and let The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation be the Hamilton–Jacobi–Bellman equation with this Hamiltonian $H$. The notions below are relative to $H$; we speak of viscosity sub- and supersolutions of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$. Upper and lower semicontinuity, local maxima and local minima refer to the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$, relative to $\mathcal{L}_{d}$.

1. (Subsolutions)¶ A map $u:\mathcal{L}_{d}\to\mathbb{R}$ is a viscosity subsolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation if it is bounded and upper semicontinuous on $\mathcal{L}_{d}$ and, for every $F\in C^{1}(E_{d})$ and every $\lambda\in\mathcal{L}_{d}$ at which $u-F\circ\iota_{d}$ has a local maximum relative to $\mathcal{L}_{d}$,

$$\rho\,u(\lambda)+H\bigl(\lambda,DF(\iota_{d}(\lambda))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF(\iota_{d}(\lambda)),\widehat{\Theta}\lambda\bigr\rangle_{d}\le f(\lambda).$$

2. (Supersolutions)¶ A map $v:\mathcal{L}_{d}\to\mathbb{R}$ is a viscosity supersolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation if it is bounded and lower semicontinuous on $\mathcal{L}_{d}$ and, for every $F\in C^{1}(E_{d})$ and every $\lambda\in\mathcal{L}_{d}$ at which $v-F\circ\iota_{d}$ has a local minimum relative to $\mathcal{L}_{d}$,

$$\rho\,v(\lambda)+H\bigl(\lambda,DF(\iota_{d}(\lambda))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF(\iota_{d}(\lambda)),\widehat{\Theta}\lambda\bigr\rangle_{d}\ge f(\lambda).$$

3. (Solutions)¶ A viscosity solution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation is a map $\mathcal{L}_{d}\to\mathbb{R}$ that is both a viscosity subsolution and a viscosity supersolution.