The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes

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

1. (The cut-off data)¶ For $x\in H$, the family $\Pi_{N}x$ of The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates lies in $H^{-1}$, which is $D(A)$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §domain and so is contained in $V$ by Hilbert Triples: Standing Notation and Background §operator, and $V$ is contained in $H$ (Hilbert Triples: Standing Notation and Background §triple); thus $\Pi_{N}$ maps $H$ into $V$. The sequence $f^{N}=(f^{N}_{j})_{j\in\mathbb{N}}$ has $f^{N}_{j}=e_{\kappa(j)}$ if $\kappa(j)\in\Gamma_{N}$ and $f^{N}_{j}=0_{H}$ otherwise, with $\kappa$ the enumeration of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise; it is square-summable in $V$, by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, because $|f^{N}_{j}|_{V}^{2}\le|e_{\kappa(j)}|_{V}^{2}$ for every $j$ and $(e_{\kappa(j)})_{j\in\mathbb{N}}$ is square-summable in $V$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise. $G_{N}:H\times H\to\mathbb{R}$ is the map $G_{N}(p,p')=\sum_{k\in\Gamma_{N}}\langle p,e_{k}\rangle_{H}\langle p',e_{k}\rangle_{H}$, a sum over the finite set $\Gamma_{N}$. $B^{N}_{q}:V\to H$ sends $x$ to $\Pi_{N}(B_{q}x)$, the family $k\mapsto2q_{k}x(k)$ on $\Gamma_{N}$ and $0$ outside, with $B_{q}$ as in The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §riccati.

2. (The operator)¶ The cut-off shifted operator at cutoff $N$ for $g$ is the function $F^{\sharp}_{N}:D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)\to\mathbb{R}$,

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

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 $B^{N}_{q}x\in H$ and $g(\Pi_{N}x)\in\mathbb{R}$; the trace $\mathrm{Tr}_{f^{N}}X$ is a real number because $f^{N}$ is square-summable in $V$; and $G_{N}(p,p)$ is a finite sum of real numbers. It is a second-order equation operator on $H$ relative to $(H,V,A)$.

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

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

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