On pg 61 of Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams, a proof of Ito's formula for continuous semimartingales involves demonstrating that the space of functions for which it applies is a multiplicative algebra. This derivation appears:
The integration by parts formula tells us that [for $X_t = X_0+M+A$ a continuous semimartingale with its canonical decomposition and $f,g$ two $C^2(\mathbb{R})$ functions for which Ito's formula holds (with $F_t := f(X_t),\ G_t := g(X_t)$),]
\begin{align}
F_tG_t-F_0G_0 &= \int_0^tF_sdG_s+\int_0^tG_sdF_s + \int_0^t f'g'(X_s)d[M]_s \\
&= \int_0^t \{ F_sg'(X_s) + f'(X_s)G_s\} dX_s + \\
&\qquad \frac{1}{2}\int_0^t\{F_sg''(X_s) + 2f'g'(X_s)+f''(X_s)G_s\}d[M]_s.
\end{align}
The first inequality is a direct application of the integration-by-parts formula, plus a recently shown identity for the third term.
The second equality applies Ito's formula to terms $\int_0^tF_sdG_s$ and $\int_0^t G_sdF_s$. But then I expect the last equality to include a $4f'g'$ term instead, since I believe
\begin{align}
\int_0^t F_s dG_s &\color{red}{=} \left[\int_0^tF_s dG_s\right]_{t=0} + \int_0^tF_sg'(X_s)d(M_s+A_s) + \frac{1}{2}\int_0^t (f'g'+fg'')(X_s)d[M_s] \\
&\color{red}{=} 0+\int_0^tF_sg'(X_s)dX_s + \frac{1}{2}\int_0^t f'g'(X_s)+F_sg''(X_s)d[M_s]. \end{align}
whence an $f'g'$ comes from each of $\int F_s dG_s$ and $\int G_s dF_s$, and another $2f'g'$ comes from the third summand.
Where am I going wrong?
Best Answer
The first red $\color{red}{=}$ is incorrect. The chain rule application there is what would be expected from classical calculus and is meaningless in this context.
The correct manipulation is more explicit with dots: \begin{align} \int_0^t F_s dG_s = F\cdot G \ |_t &= F\cdot g(X) \quad|_{t} \\ &\overset{(2)}= F\cdot (g(X_0)+g'(X)\cdot (M+A)+\frac{1}{2} g''(X)\cdot [M]) \quad|_{t} \\ &\overset{(3)}= F\cdot (g'(X)\cdot (M+A)) + F\cdot\left(\frac{1}{2}g''(X)\cdot [M]\right) \quad|_t \\ &\overset{(4)} = (Fg'(X))\cdot (M+A) + \frac{1}{2}F g''(X)\cdot [M] \quad|_t \end{align}
The first line uses definitions. (2) uses what is assumed of $g$. (3) uses distributivity of Ito integrals. (4) uses associativity of Ito integrals on both terms.