The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two

In the setting of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation, let $g:V\to\mathbb{R}$ be a running cost. The set $H$ is open in $H$ by Hilbert Triples: Standing Notation and Background §open-sets, and with $U=H$ there one has $W=D(A)$.

¶The shifted operator for $g$ is the function $F^{\sharp}:D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)\to\mathbb{R}$,

$$F^{\sharp}(x,r,p,X)=\gamma\,r-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}X+\tfrac12\,G(p,p)+\langle Ax+B_{q}x,p\rangle_{H}-g(x).$$

It is defined: for $x\in D(A)$ one has $x\in V$ and $Ax\in H$ (Hilbert Triples: Standing Notation and Background §operator), so $g(x)\in\mathbb{R}$ and $B_{q}x\in H$ (The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §riccati); $\mathrm{Tr}_{f}X\in\mathbb{R}$ because $f$ is square-summable in $V$ and $X\in\mathrm{Sym}(V)$ (The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise); and $G(p,p)\in\mathbb{R}$ because $G\in\mathrm{Sym}(H)$ and $p\in H$. It is a second-order equation operator on $H$ relative to $(H,V,A)$.

¶The shifted Hamilton-Jacobi-Bellman equation of the Wick-square problem for a function $w:H\to\mathbb{R}$ that is bounded above and below near each point of $H$ is the requirement that $w$ be a viscosity solution of $F^{\sharp}$ on $H$. For $w\in C^{2}(H)$ it is the viscosity form of

$$\gamma\,w(x)-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}\bigl(D^{2}w(x)|_{V}\bigr)+\tfrac12\,G\bigl(Dw(x),Dw(x)\bigr)+\langle Ax+B_{q}x,Dw(x)\rangle_{H}=g(x)\qquad(x\in D(A)),$$

with the restriction $D^{2}w(x)|_{V}$.