Every place I've seen defines the Ito formula for the Stratonovich integral as $df(X_t) = f'(X_t) \circ dX_t$ for $f \in C^3(\mathbb{R})$ and $X_t$ brownian motion, while the Ito integral only requires $f \in C^2$. Why is this so? More explicitly, what is wrong with the following argument: Let $f \in C^2$ , $f \notin C^3$
$$f(X_t) \circ dX_t = f(X_t) dX_t + <df'(X_t),dX_t>$$
Thus, we'd like to say that $df'(X_t) = f''(X_t) dX_t + B.V.$ term. If we had this, then we'd recover the Ito formula.
$f \in C^2$ implies $f' \in C^1$. Choose a sequence $g_n \in C^\infty$ which converge to $f'$ in $C^1$-norm.
Then, by Ito's formula,
$$dg_n(X_t) = g'_n (X_t) dX_t + \frac{1}{2}g''_n (X_t) dt$$
Now, the left hand side converges to $df'(X_t)$ and the first term on the right converges to $f''(X_t)dX_t$. Thus, the final term must also converge to some process, which we shall $dL_t$.
Now, since $\frac{1}{2}g''_n (X_t) dt = \frac{1}{2} g''^{+}(X_t) - \frac{1}{2}g''^{-}(X_t)$ where the $+$ and $-$ denote the positive and negative parts respectively, then it's clear that each element in the sequence can be written as the difference of increasing functions. Since monotonicity is preserved under limits, then $dL_t$ must also be represented in that form, hence it is of (locally) bounded variation.
What did I do wrong? Thanks for your help!
