How should $\int_0^1 |dX_s|$ be understood for a real valued semimartingale $(X_t)_{t \geq 0}$ of finite variation?

Question

How should $\int_0^1 |dX_s|$ be understood for a real valued semimartingale $(X_t)_{t \geq 0}$ of finite variation?

I read this in many sources but I can not find any explanation of this term. I know the stochastic integral but it confuses me that we take the absolute value of $dX_s$.

This notation appears in Theorem 2.3 in this paper: Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307. MR1617049

Answer 1

Suppose that $X_t$ is a continuous semi-martingale; then $X_t$ has the decomposition $$X_t = X_0 + M_t + A_t$$ where $M_t$ is a local martingale and $A_t$ is a process of bounded variation. If $X_t$ has finite variation then $M_t = 0$ (cf. Continuous local martingale of finite variation is constant). Thus, we may interpret this integral pathwise; that is, for each $\omega \in \Omega$ $$\int_0^1 |dX_s|(\omega) = \int_0^1 |dA_s|(\omega) = TV(A)(\omega)$$ where $|dA_s|$ is the total variation measure associated with $A_t$ and $TV(A)(\omega)$ is the total variation of $A_t(\omega)$.