In Shreeve's book on finance in continuous time, he "defines" the following. He says on p. 99:
In general, to compute the first-order variation of a function up to time $T$, we first choose a partition $\Pi= \{t_0, t_1, \ldots, t_n\}$ of $[0,T]$, which is a set of times $0=t_0 < t_1 < \cdots < t_n = T$. These will serve to determine the step size. We do not require the partition points $t_0 = 0, t_1, t_2, \ldots , t_n=T$ to be equally spaced, although they are allowed to be. The maximum step size of the partition will be denoted $||\Pi||= \max_{j=0, \ldots ,n-1} (t_{j+1} - t_j)$. We then define $$ FV_T (f) = \lim_{||\Pi||\to 0} \sum_{j=0}^{n-1} | f(t_{j+1}) - f(t_j)|.$$ The limit here is taken as the number $n$ of partition points goes to infinity and the length of the longest subinterval $t_{j+1} - t_j $ goes to zero.
Unfortunately, these symbols (or rather their mathematical meaning) are not defined. It is unclear to me what this "$\lim_{||\Pi||\to 0}$" means. This is not a term defined anywhere in math. Also, given this "definition" (I put it in brackets because it really isn't a definition, as it doesn't define anything unambiguously), it is unclear why this would even exist.
Can someone give me a proper definition?
Further on, he goes on to "define" the notion of 'quadratic variation':
Definition. Let $f(t)$ be a function defined for $0 \leq t \leq T$. The quadratic variation of $f$ up to time $T$ is $$ [f,f] (T) = \lim_{||\Pi||\to 0} \sum_{j=0}^{n-1} ( f(t_{j+1}) - f(t_j))^2$$ where $\Pi= \{t_0, t_1, \ldots, t_n\}$ and $0=t_0 < t_1 < \cdots < t_n = T$.
Again, the meaning of the symbol is not defined (or defined properly), and I do not know what it means.
Thanks a lot in advance for clarifying.
The limit as the mesh size goes to zero is quite standard when treating Riemann-Stieltjes integrals. The formal definition of $\lim_{\| \Pi \| \to 0}$ is the following:
Let $\Pi$ a partition of an interval $[a,b]$. and $\|\Pi\|$ as in the OP.
a sequence $X$ is called an evaluation sequence for a partition $P=\{a=:t_0< t_1 \cdots < t_n:=b\}$ if $X=\{x_1< \cdots <x_n\}$ with each $x_i \in [t_{j-1},t_j]$
Let $I(\Pi,X): \mathcal{P} \to \mathbb{R}$ a function (where $\mathcal{P}$ is the set of all the possible partitions over $[a,b]$). $$ \lim_{\|\Pi\| \to 0} I(\Pi,X)=L $$ if for each $\epsilon>0$ there exists $\delta>0$ such that, for any $\Pi \in \mathcal{P}$ such that $\|\Pi\|< \delta$ and for any evaluation sequence of $\Pi$ we have $$ |I(\Pi,X)-L| < \epsilon $$