such that $(\tau-r)$ is small enough we have by Lemmas \[lemmaphi\] and \[lemmag\] $$\frac{v(\tau,x_{t,x}(\tau)) - v(r,y)}{\tau-r} \leq
\frac{g(\tau,x_{t,x}(\tau)) - g(r,y) }{\tau-r} +
\frac{\varphi(\tau,x_{t,x}(\tau)) - \varphi(r,y)}{\tau-r}$$ $$\begin{gathered}
\leq g_t(r,y) + \frac{\int_r^\tau {\left\langle}D g(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r}
\\
+ \varphi_t(r,y) + \frac{\int_r^\tau {\left\langle}D\varphi(r,y) ,
b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r}+ {\left\langle}A^*D\varphi(r,y),y{\right\rangle}+ o(1).\end{gathered}$$ In view of the choice of $r$ we know that $$\frac{\int_r^\tau {\left\langle}D g(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r}
\xrightarrow{\tau\to r} {\left\langle}D g(r,y) , b(r,y,u(r)) {\right\rangle}$$ and $$\frac{\int_r^\tau {\left\langle}D \varphi(r,y) , b(r,y,u(s)) {\right\rangle}{\mathrm{d}}s}{\tau-r} \xrightarrow{\tau\to r} {\left\langle}D \varphi(r,y) , b(r,y,u(r))
{\right\rangle}.$$ Therefore for almost every $r$ in $[t,T]$ we have $$\begin{gathered}
\limsup_{\tau\downarrow r} \frac{v(\tau,x_{t,x}(\tau))
- v(r,x_{t,x}(r)))}{\tau-r}\\
\leq {\left\langle}D g(r,x_{t,x}(r)) + D \varphi(r,x_{t,x}(r)),
b(r,x_{t,x}(r),u(r)){\right\rangle}\\
+ g_t(r,x_{t,x}(r))+ \varphi_t(r,x_{t,x}(r)) +{\left\langle}A^* D
\varphi(r,x_{t,x}(r)),x_{t,x}(r){\right\rangle}\\
= {\left\langle}p_1(r)+p_2(r), b(r,x_{t,x}(r),u(r)){\right\rangle}+ q(r) + {\left\langle}A^*
p_1(r),x_{t,x}(r){\right\rangle}.\end{gathered}$$ We can then use Lemma \[lemmaYZ\] and (\[condmin\]) to obtain $$\begin{gathered}
\label{acbd} v(T,x_{t,x}(T)) - v(t,x)
\\
\leq \int_t^T ({\left\langle}p(r), b(r,x_{t,x}(r),u(r)){\right\rangle}+ q(r) + {\left\langle}A^*
 p_1(r),x_{t,x}(r){\right\rangle}){\mathrm{d}}r
\\
\leq \int_t^T -L(r,x_{t,x}(r),u(r)) {\mathrm{d}}r.\end{gathered}$$ Thus, using (a), we finally arrive at $$\begin{gathered}
V(T,x_{t,x}(T)) - V(t,x) = h(x_{t,x}(T)) - V(t,x) \leq h(x_{t,x}(T))
- v(t,x)
\\
= v(T,x_{t,x}(T))-v(t,x) \leq \int_t^T -L(r,x_{t,x}(r),u(r)) {\mathrm{d}}r\end{gathered}$$ which implies that $(x_{t,x}(\cdot), u(\cdot))$ is an optimal pair and that $v(t,x)=V(t,x)$.

\[remliyo\] In the book [@LiYong] (page 263, Theorem 5.5) the authors present a verification theorem 