Does Uniformly parabolic PDE has always a strong solution?

Question

Is it always true that a uniformly parabolic PDE has a strong solution in $\mathcal{C}^{1,2}([0,T)\times (a,b);\mathbb{R})$? My example is the following:
$g_t(t,y)+ \mu(t,y)g_y(t,y)+\frac{1}{2}\beta^2y^2g_{yy}(t,y)-q(t) g(t,y) + f(t,y)=0 \quad (t,y) \in [0,T), \times (a,b)$.
with boundary conditions $\begin{cases} g(t,0)=F, t \in [0,T] \\ g(T,y)= \max\{F-Ay,S\} \end{cases}$
where the functions $\mu,f,q$ are continuous and bounded. I know that I have a viscosity solution and that this solution is unique. How can I conclude that this solution is in $\mathcal{C}^{1,2}([0,T)\times (a,b);\mathbb{R})$?