Proof of Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall on Noncommutative Laws

Each result cited is applied with the data of its own statement.

Step 1 (The structure condition). By The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §data, $\mathcal{H}^{\mathrm{LQ}}$ is the linear-quadratic Hamiltonian built from $f$ and $(b_{\mu})$. Take $\rho$ from The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters, a real number with $\rho>0$ as The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws requires; the Hamiltonian of The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws §hamiltonian does not involve $\rho$. The hypotheses of The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure are then exactly those assumed here: the same bounds and Lipschitz estimates with constants $a\ge0$ and $L\ge0$ on $A(\mu)_{ij}$ and $c(\mu)_{i}$, and $f$ bounded and uniformly continuous on $\Sigma^{2}_{d}$ with the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics. Hence, by The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure §structure, $\mathcal{H}^{\mathrm{LQ}}$ satisfies the structure condition.

Step 2 (Comparison and uniqueness). Take the Hamiltonian $\mathcal{H}$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian to be $\mathcal{H}^{\mathrm{LQ}}:\Sigma^{2}_{2d}\to\mathbb{R}$. By The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §equation, $(\mathrm{E}^{\mathrm{LQ}})$ is the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §score-form with this Hamiltonian, and by the preamble of Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall free-energy-penalised viscosity subsolutions, supersolutions and solutions of $(\mathrm{E}^{\mathrm{LQ}})$ are those of $(\mathrm{E})$ with $\mathcal{H}=\mathcal{H}^{\mathrm{LQ}}$. By Step 1 this Hamiltonian satisfies the structure condition, so Comparison and Uniqueness for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall applies. Clause 1 is then Comparison and Uniqueness for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §comparison, applied to the given $u$ and $v$, and clause 2 is Comparison and Uniqueness for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §uniqueness, applied to any two bounded functions $\Sigma^{2}_{d}\to\mathbb{R}$ that are weak-star continuous on bounded laws and free-energy-penalised viscosity solutions of $(\mathrm{E}^{\mathrm{LQ}})$.