the small differences in the definition of test2 functions) from Theorem 2.2, page 229 of [@LiYong] and the proof of Theorem 3.2, page 240 of [@LiYong] (or from [@CL5]).

We will need a comparison result in the proof of the verification theorem. There are various versions of such results for equation (\[deterministicHJB\]) available in the literature, several sufficient sets of hypotheses can be found in [@CL4; @CL5]. Since we are not interested in the comparison result itself we choose to assume a form of comparison theorem as a hypothesis.

\[D4deterministiccomparison\] There exists a set $\mathcal{G}\subseteq C([0,T]\times\mathcal{H})$ such that:

-   the value function $V$ is in $\mathcal{G}$;

-   if $v_1, v_2 \in \mathcal{G}$, $v_1$ is a subsolution of the HJB equation (\[deterministicHJB\]) and $v_2$ is a supersolution of the HJB equation (\[deterministicHJB\]) then $v_1\leq v_2$.

Note that from $(i)$ and $(ii)$ we know that $V$ is the only solution of the HJB equation (\[deterministicHJB\]) in $\mathcal{G}$.

We will use the following lemma whose proof can be found in [@YongZhou], page 270.

\[lemmaYZ\] Let $g\in C([0,T];\mathbb{R})$. We extend $g$ to a function (still denoted by $g$) on $(-\infty,+\infty)$ by setting $g(t)=g(T)$ for $t>T$ and $g(t)=g(0)$ for $t<0$. Suppose there is a function 