\in test1$ and $g\in
test2$, we have $$\label{eqsupersol} -\varphi_t(\bar t, \bar x) - g_t(\bar t, \bar x)
- {\left\langle}A^* D \varphi(\bar t, \bar x) , \bar x {\right\rangle}+ H(\bar t, \bar
x, -D \varphi(\bar t, \bar x)- D g(\bar t, \bar x)) \leq 0.$$

\[defdeterministicsol\] A function $v\in
C((0,T]\times\mathcal{H})$ is a (viscosity) *solution* of the HJB equation (\[deterministicHJB\]) if it is at the same time a subsolution and a supersolution.

We will be also using viscosity sub- and supersolutions in situations where no terminal values are given in (\[deterministicHJB\]). We will then call a viscosity subsolution (respectively, supersolution) simply a function that satisfies (\[eqsubsol\]) (respectively, (\[eqsupersol\])).

\[lemmaphi\] Let Hypotheses \[hpD2onb\] and \[hpD3onLandh\] hold. Let $\phi\in test1$ and $(t,x)\in(0,T)\times\mathcal{H}$. Then the following convergence holds uniformly in $u(\cdot) \in
\mathcal{U}[t,T]$: $$\begin{gathered}
\lim_{s\downarrow t} \left ( \frac{1}{s-t} \left (
\varphi(s,x_{t,x}(s)) - \varphi(t,x) \right ) - \varphi_t(t,x) -
{\left\langle}A^*D\varphi (t,x),x{\right\rangle}\right.\\
\left. - \frac{1}{s-t} \int_t^s {\left\langle}D \varphi (t,x), b(t,x,u(r))
{\right\rangle}{\mathrm{d}}r \right ) =0\end{gathered}$$ Moreover we have for $s-t$ sufficiently small $$\begin{gathered}
\label{eq:explicitphi} \varphi(s,x_{t,x}(s))-\varphi(t,x) = \int_t^s
\varphi_t(r,x_{t,x}(r)) +
{\left\langle}A^*D\varphi (r,x_{t,x}(r)),x_{t,x}(r){\right\rangle}\\
+ {\left\langle}D \varphi(r,x_{t,x}(r)), b(r,x_{t,x}(r),u(r)) {\right\rangle}{\mathrm{d}}r\end{gathered}$$

See [@LiYong] Lemma 3.3 page 240 and Proposition 5.5 page 67.

\[lemmag\] Let Hypotheses \[hpD2onb\] and \[hpD3onLandh\] hold. Let $g\in test2$ 