Bounded variation and continuous local part when using Ito's Formula

Question

When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the case?

Answer 1

If $(X_t)_{t\geq 0}$ is a continuous semimartingale, then it has a representation of the form $$ X_t=X_0+M_t+A_t,\quad t\geq 0, $$ where $X_0$ is $\mathcal{F}_0$-measurable, $(M_t)_{t\geq 0}$ is a continuous local martingale and $(A_t)_{t\geq 0}$ is continuous and of bounded variation. When applying Itô's formula, the continuous local martingale part is everything that involves integration with respect to $(M_t)_{t\geq 0}$ and the bounded variation part is the rest.

In other words, if $f$ is a $C^2(\mathbb{R})$ function, then $(f(X_t))_{t\geq 0}$ and $(f(X_0)+M^f_t+A^f_t)_{t\geq 0}$ are indistinguishable, where $$ M^f_t=\int_0^t\frac{\partial}{\partial x}f(X_s)\,\mathrm dM_s,\quad t\geq 0 $$ is the continuous locale martingale part and $$ A^f_t=\int_0^t\frac{\partial}{\partial x}f(X_s)\,\mathrm d A_s+\frac{1}{2}\int_0^t\frac{\partial^2}{\partial x^2}f(X_s)\,\mathrm d [X]_s,\quad t\geq 0 $$ is the bounded variation part. I'm sure you can extend this to the case where $f\in C^2(\mathbb{R}^n)$.