# Does Itô's formula define a semimartingale?

We may define a semimartingale as: $$X_{t}=X_{0}+M_{t}+A_{t}$$ Where $$X_0$$ is $$\mathcal{F}_0$$ measurable, $$M$$ is a continuous local martingale $$M_0 = 0$$ and $$A$$ is an adapted continuous finite variation process with $$A_0 = 0$$. The Itô's formula reads: $$df(t,X_t) =\frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial X_s}\,dX_t+\frac{1}{2}\frac{\partial^2f}{\partial X_s^2}d[X]_t$$

$$[X]_t$$ is a quadratic variation and hence a finite variation process, is $$\frac{\partial^2f}{\partial x^2}d[X]_t$$ also a finite variation process?

• @Calculon I am not sure what you mean. The first ($dt$, which is missing) and last ($d[X]_t$) parts of the formula give finite variation processes, while the $dX_t$ part is a semimartingale. This is the nice thing about semimartingales, they can be integrated, and are stable by composition. Commented Jan 3, 2021 at 11:22

Since $$[X]_t$$ is of finite variation, $$\int f_{xx}(s, X_s)d[X]_s$$ is interpreted in the Lebesgue-Stieltjes sense. This integral itself produces finite variation functions. You can see that by decomposing $$f_{xx}(s, X_s)$$ into positive and negative parts and $$d[X]_s$$ as the difference of two nondecreasing functions.