Quadratic variation along a partition sequence

Fix $T>0$ and a continuous function $x:[0,T]\to \mathbb{R}$ (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point). A partition sequence of $[0,T]$ is a sequence $({\pi }_n{)}_{n\ge 0}$ of partitions of $[0,T]$ in the sense of Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, written ${\pi }_n=(m_n,s^{(n)}),0=s_0^{(n)}<s_1^{(n)}<\cdots <s_{m_n}^{(n)}=T,$ whose mesh $\mathrm{mesh}({\pi }_n):={\max }_{1\le k\le m_n}(s_k^{(n)}-s_{k-1}^{(n)})$ tends to zero as $n\to \mathrm{\infty }$. The partitions need not refine one another and no regularity of the points beyond mesh convergence is assumed. A family originally indexed by positive integers is read as the zero-indexed family $q_n={\pi }_{n+1}$; this changes none of its limiting assertions.

For $t\in [0,T]$ two partial sums are attached to $({\pi }_n)$. If $t$ is a partition point, both are defined by the same formula; the cases $t=0$ and $t=T$ are included, and an empty sum is $0$.

1. Step convention. With $k(t)$ the largest index in $\{0,\dots ,m_n\}$ with $s_{k(t)}^{(n)}\le t$, $[x{]}_t^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}:={\sum }_{k=1}^{k(t)}(x_{s_k^{(n)}}-x_{s_{k-1}^{(n)}}{)}^2\mathrm{.}$
2. Partial-increment convention. On the interval $[s_{k(t)}^{(n)},s_{k(t)+1}^{(n)}]$ containing $t$ one also adds the terminal increment, $[x{]}_t^{{\pi }_n,\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}:=[x{]}_t^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}+(x_t-x_{s_{k(t)}^{(n)}}{)}^2(k(t)<m_n),$ and $[x{]}_T^{{\pi }_n,\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}:=[x{]}_T^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}$.

The quadratic variation of $x$ along $({\pi }_n)$ is the limit of either family of functions, in a mode that is part of every later statement: the pointwise claim is that $[x{]}_t^{{\pi }_n}$ converges as $n\to \mathrm{\infty }$ for each fixed $t\in [0,T]$, and the uniform claim is that it converges uniformly on $[0,T]$. No other mode and no other partition family is included.

Three conventions are built into the definition and are used later in this form.

1. The two conventions differ by at most the squared maximal oscillation. For every $n$ and $t$, $\mid [x{]}_t^{{\pi }_n,\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}-[x{]}_t^{{\pi }_n,\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}\mid =(x_t-x_{s_{k(t)}^{(n)}}{)}^2\le ({\max }_k{\sup }_{u,v\in [s_{k-1}^{(n)},s_k^{(n)}]}\mathrm{\mid }{x}_u-x_v\mathrm{\mid }{)}^2,$ and the right-hand side tends to $0$ as $n\to \mathrm{\infty }$ because $x$ is uniformly continuous on the compact interval $[0,T]$ and the mesh tends to $0$. For completeness this uniform-continuity assertion is choice-free: for each $c\in [0,T]$ and $\epsilon >0$, continuity supplies a least integer $j(c)\ge 0$ such that $\mathrm{\mid }{x}_y-x_c\mathrm{\mid }<\epsilon \mathrm{/}2$ whenever $y\in [0,T]$ and $\mathrm{\mid }y-c\mathrm{\mid }<2^{1-j(c)}$. The intervals of radius $r_c=2^{-j(c)}$ centered at $c$ cover $[0,T]$. Its compactness Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact gives finitely many such intervals covering it. Put $\delta$ equal to the minimum of their positive radii. If $\mathrm{\mid }u-v\mathrm{\mid }<\delta$ and $u$ belongs to the interval centered at $c$, then both $u,v$ are within $2r_c$ of $c$, so $\mathrm{\mid }{x}_u-x_v\mathrm{\mid }<\epsilon$. This proves uniform continuity without selecting arbitrary radii. In particular the two conventions have the same limit whenever either limit exists.
2. No partition-independent object is defined. The symbol $[x{]}^{{\pi }_n}$ names the $n$th sum along the named sequence $({\pi }_n)$, and any quadratic-variation limit is attached to that sequence; it is not a claim that the sums converge along every refining sequence, nor that a path-dependent choice of partitions leaves the limit unchanged. When a statement below says "quadratic variation", the partition sequence is part of the data.
3. Dependence on $t$. Each $[x{]}_t^{{\pi }_n}$ is a genuine real number, being a finite sum of nonnegative terms; the family $t\mapsto [x{]}_t^{{\pi }_n}$ is nondecreasing in $t$ for the step convention, and for the partial-increment convention it agrees with the step value at partition points.

No choice principle is used: the partitions are given as a sequence of finite lists, the sums are finite sums of real numbers, and the limits are the usual uniqueness-of-limit limits.

Lawler, Section 2.8, defines the quadratic sums along a partition sequence, proves the convergence results for meshes tending to zero (Theorems 2.8.1-2.8.2) and warns explicitly that the mesh condition is load-bearing: without a prescribed partition family the sums may depend on the partitions chosen. The definition above separates the two partial-sum conventions used in that section and records only a difference bound, so that later items can state their limits for either convention without redefining the symbol.