Can we apply Ito formula to quadratic variation of $C^1$ function of semimartingale?

Question

I'm looking at conversion between Ito and Stratonovich for certain classes of functions, and was wondering whether we can apply Ito's to a $C^1$ function in a covariation expression: Ito's applies normally only to $C^2$ function $F(X)$ giving $$ dX_t = F'(X_t)dX_t + \dfrac{1}{2} F''(X_t) d \langle X \rangle_t $$ which for quadratic co-variation with some other semimartingale $W$ becomes (where the finite variation term disappears) (integrating from $0$ to $t$) \begin{align} \langle X,W \rangle &= \langle \int dX_t + W \rangle \\ &= \langle \int F'(X_s)\, dX_s,W\rangle \\ &= \int F'(X_s) \, d\langle X,W\rangle \end{align} and I'm wondering whether since we only have the expression of $F'$ if this can be done more generally with $C^1$ functions, or if perhaps the "hypothetical" 2nd order term in any approximation (for example a Stone-Weierstrass polynomial one approximating $F$) is too large in the limit.

Answer 1

Convex case

A very general formula is the The Itô-Tanaka-Meyer Formula (here too "Itô–Tanaka formula and local times of semimartingales")

Theorem 2 (Itô-Tanaka-Meyer) Let X be a semimartingale, $f\colon{\mathbb R}\rightarrow{\mathbb R}$ be convex, and $L^x_t$ be a jointly measurable version of the local times. Then,

$\begin{aligned} f(X_t)=& f(X_0)+\int_0^t f^\prime(X_-)dX+\frac12\int_{-\infty}^\infty L^x_t\,f^{\prime\prime}(dx)\\ &\quad+\sum_{s\le t}\left(\Delta f(X_s)-f^\prime(X_{s-})\Delta X_s\right) \end{aligned}$

almost surely, for each $t\ge0$.

See here "Itô's Formula for Non-Smooth Functions" for the case of generalized first/second derivatives. Here too Other versions of a weak Ito formula?

$C^1$ and Martingale

As mentioned here Applying Ito's formula on a $C^1$ only differentiable function yielding a martingale

Assume that we know that $f(s+t,x+B_t)_{t \geq 0}$ is a martingale for any fixed $s \geq 0$ and $x \in \mathbb{R}$, and assume that $f$ grows at most sub-exponentially, we can apply the martingale property and Itô's formula to deduce that

$$f(t,B_t) = f(0,B_0) + \int_0^t f'(s,B_s) \, dB_s.$$

$C^1$ extensions

In the works

• "Ito formula for cl-functions of semimartingales"
• "A $C^{ 0,1}$-functional Itˆo’s formula and its applications in mathematical finance"
• "Extended Itô calculus for symmetric Markov processes",

they obtain some generalized formulas for some specific settings.

Fukushima decomposition

As mentioned in "Ito formula for cl-functions of semimartingales", we have that if $f \in C^1(R)$ and $X$ is semimartingale, then $f(X)$ is a Dirichlet process (i.e. the sum of a martingale and a process with zero-energy) and is not in general a semimartingale (they reference an article in French, but I would like to write the counterexample down once I understand it.).