---
title: Pathwise Quadratic Variation Terms
url: https://www.emergentmind.com/topics/pathwise-quadratic-variation-terms
type: topic
---

# Pathwise Quadratic Variation Terms

Pathwise quadratic variation terms are fundamental objects in stochastic analysis and model-free mathematical finance, central to any “probability-free” Itô calculus for irregular paths, including those with jumps. The rigorous construction and canonical properties of these terms are established for d-dimensional càdlàg paths with upward-unrestricted and downward-moderately restricted jumps, constructed over arbitrary sequences of “optional” (i.e., stopping-time) partitions or via partition-free functionals related to truncated variation. This enables the pathwise development of integration, Itô-type formulas, and robust analysis of price dynamics in a non-probabilistic framework.

## 1. Sample Path Class and Mild Downward-Jump Constraint

A pathwise quadratic variation theory for model-free price paths requires structural assumptions on admissible paths. Consider a fixed time interval $[0,T]$, and let $v:\mathbb{R}_+\to\mathbb{R}_+$ be any non-decreasing function. Work on the path space
\[
Q \subset D([0,T],\mathbb{R}^d)
\]
of all càdlàg paths $w = (w^1,\ldots,w^d)$ such that, for each $i=1,\ldots,d$ and $t\in(0,T]$,
\[
\Delta^-w^i(t) := w^i(t) - w^i(t-) \ge -v\big(\sup_{0\le s < t}|w(s)|\big).
\]
Arbitrary upward jumps are permitted; only downward jumps are controlled by a function of the past path supremum. The pathwise filtration is the raw (canonical) filtration generated by the coordinate processes, universally completed.

## 2. Optional Partitions and Quadratic Variation Limit

An optional partition $T$ of $[0,T]$ is a (possibly random) non-decreasing sequence of stopping times $0 = T_0 \le T_1 \le \cdots \le T_{N_n} = T < +\infty$, with $N_n \in \mathbb{N}\cup\{\infty\}$ and $T_k \equiv T$ from some index onward. The mesh-oscillation for a path $w\in Q$ along $T$ is
\[
\operatorname{Osc}(w, T) = \max_{k=1,\ldots, N_n} \sup_{s, t\in [T_{k-1}, T_k)} |w(t)-w(s)|.
\]
Given a sequence of nested optional partitions $(T^n)$, define the discrete quadratic variation process
\[
Q^n(w; t) := \sum_{k : T^n_k < t} |w(T^n_{k+1}) - w(T^n_k)|^2, \qquad t \in [0,T].
\]
The pathwise quadratic variation $[w, w](t)$ is defined as the uniform limit of $Q^n(w; t)$ as $n\to\infty$:
\[
Q^n(w; \cdot) \xrightarrow[]{\text{unif}} [w, w](\cdot).
\]

## 3. Partition-Independence and Cross-Variation

A crucial property is partition-independence: for $w \in Q_{q, M}$ (i.e., paths with $\sup_t|w(t)| \leq M$ and bounded quadratic variation along canonical partitions), if $\operatorname{Osc}(w, T^n) \to 0$ "in outer measure," then $Q^n(w; \cdot) \to [w, w](\cdot)$ in the same sense, and this limit does not depend on the choice of $(T^n)$. In particular, partition-independence holds for any family of stopping-time partitions whose mesh-oscillation vanishes in the specified sense. Polarization extends the result to the cross-variations $[w^i, w^j](t)$, constructed using the standard bilinear identity
\[
[w^i, w^j](t) = \frac{1}{2}\left([w^i + w^j](t) - [w^i](t) - [w^j](t)\right).
\]
The key proof steps involve an integration-by-parts argument to show the difference between $Q^n(w;t)$ and $[w, w](t)$ is a model-free Itô-type integral, which vanishes as oscillation diminishes [1710.07894].

## 4. Control-of-Oscillation, Outer Measure, and Typical Paths

The oscillation-control condition guarantees robustness of the theory. For every $q, M > 0$, if
\[
P[w \in Q_{q, M} : \operatorname{Osc}(w, T^n) > \varepsilon] \to 0 \quad (n \to \infty)
\]
for Vovk’s outer measure $P$, one obtains convergence in outer measure (hence quasi-surely for typical paths), ensuring the pathwise quadratic variation exists and is partition-independent for almost all paths in the game-theoretic sense [1710.07894], [1108.0799].

## 5. Partition-Free Description: Truncated Variation Functional

A partition-free, quasi-explicit characterization of the continuous part of the quadratic variation arises from the truncated variation functional. For a real càdlàg function $f : [0,T]\to\mathbb{R}$ and truncation threshold $c \geq 0$,
\[
\operatorname{TV}^c(f; [0,t]) := \sup_{\text{partitions}} \sum_{i=1}^n \max\{|f(t_i) - f(t_{i-1})| - c, 0 \}.
\]
One then obtains the continuous part of the quadratic variation via
\[
\lim_{c \to 0^+} c \cdot \operatorname{TV}^c(w^i; [0,t]) = [w^i, w^i]^{\text{cont}}(t) \qquad (i=1,\ldots, d),
\]
and, by polarization,
\[
\frac{1}{2}\lim_{c \to 0^+}\big( \operatorname{TV}^c(w^i + w^j) - \operatorname{TV}^c(w^i - w^j) \big) = [w^i, w^j]^{\text{cont}}(t).
\]
Thus, $[w, w]$ can be recovered entirely as a pathwise, partition-free limit of functionals based on truncated variation [1710.07894].

## 6. Stochastic Integration and the Pathwise Itô Formula

Once pathwise quadratic variation has been constructed, a model-free Itô integral can be defined using non-anticipative Riemann sums along sufficiently fine optional partitions. The pathwise Itô–Föllmer formula holds for $F \in C^2(\mathbb{R})$:
\[
F(w_t) = F(w_0) + \int_0^t F'(w_{s-})\, d w_s + \frac{1}{2}\int_0^t F''(w_s)\, d[w, w](s) + \sum_{0 < s \leq t}\big(F(w_s) - F(w_{s-}) - F'(w_{s-}) \Delta w_s\big).
\]
Here, $d[w, w](s)$ is the pathwise quadratic variation, formed as the uniform limit over the refining sequence [1710.07894], [1108.0799].

## 7. Summary Table of Key Quadratic Variation Terms

| Notation                 | Description                        | Construction Principle                                    |
|--------------------------|------------------------------------|----------------------------------------------------------|
| $Q^n(w;t)$               | Discrete QV along partition        | $\sum_{k: T^n_k < t} |w(T^n_{k+1}) - w(T^n_k)|^2$        |
| $[w, w](t)$              | Pathwise quadratic variation       | Uniform limit of $Q^n(w; t)$ or from truncated variation |
| $\operatorname{TV}^c(f)$ | Truncated variation functional     | Partition-free, supremum over all partitions             |
| $[w^i,w^j](t)$           | Cross variation (polarization)     | $1/2\left([w^i + w^j](t) - [w^i](t) - [w^j](t)\right)$   |
| $F(w_t)$ expansion       | Pathwise Itô–Föllmer formula       | Involves $[w, w]$ for the “Itô term”                    |

Both the optional-partition and truncated-variation formulations are fully pathwise, maintain compatibility with the classical semimartingale bracket (almost surely), and are robust to the partition-choice as long as control-of-oscillation holds [1710.07894].

## 8. Relation to Semimartingale Theory and Broader Significance

For càdlàg semimartingales with mildly restricted downward jumps, this pathwise construction recovers the standard quadratic variation almost surely, showing the framework is a strict generalization of semimartingale theory, not reliant on probabilistic structure. This approach extends the foundations of robust financial mathematics, stochastic control, and analysis of path-dependent options in environments where probabilistic modeling of the underlying process is either unavailable or undesirable. The model-free underpinnings, partition-invariance, and partition-free characterization contribute to the understanding of pathwise Itô calculus applicable in a broad range of stochastic analysis and mathematical finance contexts [1710.07894].

Source: https://www.emergentmind.com/topics/pathwise-quadratic-variation-terms