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)?
As explained at Converse of Itô's formula, such a formula would imply that $f$ is $C^2$-differentiable.
Let $f$, $h$, $g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$ Prove that $f$ is a $C^2$-function, $f'=g$ and $f''=h$.
In terms of semi-counterexample, the absolute value function is uniformly contiuous but it satisfies a formula that requires the addition of Local time (and so having an Itô formula would imply that local time is zero). Tanaka's formula:
$$|B_t| = \int_0^t \operatorname{sgn}(B_s)\, dB_s + L_t.$$
There are weaker versions that require only convexity e.g. The Itô–Tanaka–Meyer Formula.
