Let $d X_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$ be an Ito diffusion. If we choose a continuously twice twice differentiable function $f$ with compact support and define $u(t,x) = E( f(X_t) | X_0 = x)$ then by Kolmogorov's theorem $\frac{\partial u}{\partial t } = Au$, where $A$ is a generator of $X$, which in particular means that $u$ is twice differentiable with respect to $x$.
What one needs to assume additionally (about $f$ and / or $b,\sigma$ ) in order to obtain three times (or generally higher orders of) differentiability of $u$ with respect to $x$?
That is now a question about regularity of solutions of parabolic differential equations.
In a nutshell and somewhat simplified, if $b \in C^{k+1,\alpha}, \, \sigma \in C^{k,\alpha}$, then $u \in C^{k+2,\alpha}$ for $t > 0$. Here $C^{k,\alpha}$ means $k$ times differentiable with $k$-th derivatives being $\alpha$ Holder continuous. If also $f \in C^{k+2,\alpha}$, then the solution is in the same space all the way to $t = 0$. Clearly one cannot do better, so this is called "maximal regularity".
Classical references are the books by Friedman from 1969 or so and by Ladyzhenskaya from around that time.