In chapter 1 definition 2.3 of Revuz's and Yor's Continuous Martingales and Brownian Motion is defined the quadratic variation of a real-valued process $X$ as a finite process $\left(\langle X, X\rangle_t\right)_t$ such that for all $t\in\mathbf{R}_+$ and every sequence $(\pi_n)_n$ of subdivisions of $[0,t]$ such that $\|\pi_n\| \to 0$ as $n\to +\infty$ we have $T_t^{\pi_n} (X) \to \langle X, X\rangle_t$ in probability as $n\to +\infty$, where if $\pi = \{0 = t_0 < \ldots < t_k = t\}$ is a subdivision of $[0,t]$ we note $\|\pi\| = \max_{i=0,\ldots,k-1} |t_{i+1} - t_i|$ and $T_t^{\pi} (X) = \sum_{i = 0}^{k-1} \left( X_{t_{i+1}} - X_{t_i}\right)^2$.
What is intended by finite process exacty ? (It does not seem to be defined in the book.) Does this means a process $(Y_t)_{t\geq 0}$ with values in $\overline{\mathbf{R}}$ such that :
- $\forall t\in\mathbf{R}_+, Y_t \in \mathbf{R}$
- $\forall t\in\mathbf{R}_+$, $\mathbf{P}$-a.s. $Y_t \in \mathbf{R}$
- $\mathbf{P}$-a.s. $\forall t\in\mathbf{R}_+$, $Y_t \in \mathbf{R}$
I don't think it can be (1) but I am hesitating between (2) and (3) even if I would go for (2).