C^1((0,T)\times\mathcal{H}) \; :& \exists g_0, \colon
[0,+\infty) \to [0,+\infty), \;\\
&and \; \eta\in C^1((0,T)) \text{ positive } \; s.t.\\
&g_0 \in C^1([0,+\infty)), \; g_0'(r) \geq 0 \; \forall r\geq 0, \\
& g_0'(0)=0 \; and \; g(t,x)=\eta(t)g_0(\|x\|) \\
&\forall (t,x)\in (0,T)\times \mathcal{H} \}
\end{array}$$ We use test2 functions that are a little different from the ones used in [@CL4]. The extra term $\eta(\cdot)$ in test2 functions is added to deal with unbounded solutions. We recall that $D\varphi$ and $Dg$ stand for the Frechet derivatives of these functions.

\[defdeterministicsubsol\] A function $v\in
C((0,T]\times\mathcal{H})$ is a (viscosity) *subsolution* of the HJB equation (\[deterministicHJB\]) if $$v(T,x) \leq h(x) \;\;\; for \; all\; x\in\mathcal{H}$$ and whenever $v-\varphi-g$ has a local maximum at $(\bar t, \bar
x)\in[0,T)\times\mathcal{H}$ for $\varphi \in test1$ and $g\in
test2$, we have $$\label{eqsubsol} \varphi_t(\bar t, \bar x) + g_t(\bar t, \bar
x)+{\left\langle}A^* D \varphi(\bar t, \bar x) , \bar x {\right\rangle}+H(\bar t, \bar
x, D\varphi(\bar t, \bar x)+ Dg(\bar t, \bar x)) \geq 0.$$

\[defdeterministicsupersol\] A function $v\in
C((0,T]\times\mathcal{H})$ is a (viscosity) *supersolution* of the HJB equation (\[deterministicHJB\]) if $$v(T,x) \geq h(x) \;\;\; for \; all\; x\in\mathcal{H}$$ and whenever $v+\varphi+g$ has a local minimum at $(\bar t, \bar
x)\in[0,T)\times\mathcal{H}$ for $\varphi 