Pathwise Quadratic Variation Terms

• Pathwise quadratic variation terms are measures defined as the uniform limit of discrete square increments for càdlàg paths with controlled downward jumps.
• They are constructed using optional partitions and truncated variation functionals to ensure partition-independence and robustness in non-probabilistic settings.
• This approach underpins a pathwise Itô formula for stochastic integration, advancing model-free techniques in mathematical finance.

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 (Galane et al., 2017).

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 (Galane et al., 2017, Vovk, 2011).

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 (Galane et al., 2017).

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 (Galane et al., 2017, Vovk, 2011).

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 (Galane et al., 2017).

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 (Galane et al., 2017).