Consider a Brownian motion $(W_t)_{t\in[0;T]}$.

If $f\colon [0;T] \times \mathbb R \to \mathbb R$ is $C^{1,2}$, we know that $(f(t,W_t))_{t\in[0;T]}$ is an Ito process and we can directly write down the drift and volatility process.

Can we say that it is also an Ito process if $f$ is only uniformly continuous? If not, can we say this at least for weaker assumptions on $f$ (maybe without the possibility to compute the drift and volatility explicitly)?

Consider a Brownian motion $(W_t)_{t\in[0;T]}$.

If $f\colon [0;T] \times \mathbb R \to \mathbb R$ is $C^{1,2}$, we know that $(f(t,W_t))_{t\in[0;T]}$ is an Itô process and we can directly write down the drift and volatility process.

Can we say that it is also an Itô process if $f$ is only uniformly continuous? If not, can we say this at least for weaker assumptions on $f$ (maybe without the possibility to compute the drift and volatility explicitly)?