The Riccati Shift of the Galerkin Equations of the Wick-Square Problem: a Penalty-Drift Equation with Bounded Cost, and Well-Posedness at Each Cutoff

In the setting of The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential, let $C_{g}\in\mathbb{R}$ be nonnegative, let $g:H^{-1}\to\mathbb{R}$ satisfy $|g(x)|\le C_{g}$ for every $x\in H^{-1}$, and let $N\in\mathbb{N}$, with $m$, $\iota_{N}$, $\tilde{u}_{N}$ and $\Phi_{N}$ as fixed in The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates and The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §functions. Let $\mathcal{F}_{N}$ be the Wick-ordered Galerkin operator and $\mathcal{F}^{\sharp}_{N}$ the shifted Galerkin operator for $g$, and let viscosity sub- and supersolutions be those of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation. Then the following hold.

1. (The Riccati shift)¶ For every $z\in\mathbb{R}^{m}$, $D\tilde{u}_{N}(z)$ is the vector with components $2q_{\kappa_{N}(i)}z_{i}$ and $D^{2}\tilde{u}_{N}(z)$ is the diagonal matrix with diagonal entries $2q_{\kappa_{N}(i)}$; and for all $z,p\in\mathbb{R}^{m}$, $r\in\mathbb{R}$ and $X\in\mathcal{S}(m)$,

$$\mathcal{F}_{N}\bigl(z,\,r+\tilde{u}_{N}(z),\,p+D\tilde{u}_{N}(z),\,X+D^{2}\tilde{u}_{N}(z)\bigr)=\mathcal{F}^{\sharp}_{N}(z,r,p,X).$$

2. (The potential)¶ $\Phi_{N}$ is a penalty on $\mathbb{R}^{m}$, and for the penalty-drift operator with potential $\Phi_{N}$, discount $\gamma$ and noise intensity $\nu$ the conditions Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §one-sided and Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §dissipation hold, with $c_{R}=0$ for every $R$, with $\varepsilon=1$, and with $C=\tfrac{\nu}{2}\sum_{i=1}^{m}\bigl(\mu_{\kappa_{N}(i)}+2q_{\kappa_{N}(i)}\bigr)$.

3. (Solutions correspond)¶ A function $U:\mathbb{R}^{m}\to\mathbb{R}$ is a viscosity subsolution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ if and only if $U-\tilde{u}_{N}$ is a viscosity subsolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$; and likewise for viscosity supersolutions.

4. (Well-posedness at each cutoff)¶ Assume moreover that $g\circ\iota_{N}$ is continuous from $(\mathbb{R}^{m},d_{E})$ to the real line. Then there is exactly one function $U_{N}:\mathbb{R}^{m}\to\mathbb{R}$ that is a viscosity subsolution and a viscosity supersolution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ and for which $U_{N}-\tilde{u}_{N}$ is bounded. Moreover $U_{N}-\tilde{u}_{N}$ is continuous on $\mathbb{R}^{m}$ and $|U_{N}(z)-\tilde{u}_{N}(z)|\le C_{g}/\gamma$ for every $z\in\mathbb{R}^{m}$.