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Brownian quadratic variation on dyadic partitions

Statement

Let $B$ be a standard Brownian motion Brownian motion and fix $T > 0$ . For $n \geq 1$ let $π_{n}$ be the dyadic partition of $[0, T]$ with points $k T / 2^{n}$ , $k = 0, \dots, 2^{n}$ , and let $Q_{n} : = \sum_{k = 1}^{2^{n}} (B_{k T / 2^{n}} - B_{(k - 1) T / 2^{n}})^{2} = [B]_{T}^{π_{n}}$ in the notation of Quadratic variation along a partition sequence. Then $Q_{n} \to T$ in $L^{2}$ and almost surely as $n \to \infty$ .

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , $T > 0$ , and the dyadic partitions $π_{n}$ above with $h = T / 2^{n}$ .

[F1]

The increments of $B$ over disjoint intervals are independent with laws $N (0, h)$ for interval length $h$ . Brownian motion

[F2]

If $X_{t} - X_{s}$ has law $N (0, ∣ t - s ∣)$ then $E ∣ X_{t} - X_{s} ∣^{2 m} = c_{m} ∣ t - s ∣^{m}$ with $c_{m} = (2 m - 1)!!$ ; in particular $E (Δ B)^{2} = h$ and $E (Δ B)^{4} = 3 h^{2}$ for an increment of length $h$ . Gaussian even moments for Brownian increments

[F3]

Chebyshev: $P (∣ X - E X ∣ \geq λ) \leq Var (X) / λ^{2}$ for a square-integrable real $X$ and $λ > 0$ . Chebyshev's inequality for random variables

[F4]

First Borel-Cantelli: if $\sum_{n} P (G_{n}) < \infty$ then almost surely only finitely many $G_{n}$ occur. First Borel-Cantelli lemma for events

[F5]

$[B]_{T}^{π_{n}}$ denotes the terminal quadratic sum along the named partition sequence $π_{n}$ , whose mesh $T / 2^{n}$ tends to zero. Quadratic variation along a partition sequence

[F6]

AC is the ambient assumption of the Brownian and normal-law interfaces. The Axiom of Choice

Proof

technique · direct

1.1

Writing $Δ_{k} : = B_{k h} - B_{(k - 1) h}$ for $k = 1, \dots, 2^{n}$ , [F1] and [F2] give $E Δ_{k}^{2} = h$ and $E Δ_{k}^{4} = 3 h^{2}$ , so the centered variables $Y_{k} : = Δ_{k}^{2} - h$ satisfy $E Y_{k} = 0$ and $Var (Y_{k}) = E Δ_{k}^{4} - (E Δ_{k}^{2})^{2} = 3 h^{2} - h^{2} = 2 h^{2}$ .

F1F2

2.1

$Q_{n} - T = \sum_{k = 1}^{2^{n}} Y_{k}$ has mean $0$ , and [F1] makes the $Y_{k}$ independent, so $Var (Q_{n} - T) = \sum_{k} Var (Y_{k}) = 2^{n} \cdot 2 (T / 2^{n})^{2} = 2 T^{2} / 2^{n}$ ; hence $E (Q_{n} - T)^{2} = 2 T^{2} / 2^{n} \to 0$ and $Q_{n} \to T$ in $L^{2}$ .

step 1.1F1

3.1

For every $ε > 0$ , [F3] gives $P (∣ Q_{n} - T ∣ \geq ε) \leq Var (Q_{n} - T) / ε^{2} = 2 T^{2} / (2^{n} ε^{2})$ , which is summable in $n$ ; applying [F4] to $G_{n} = {∣ Q_{n} - T ∣ \geq \frac{1}{m}}$ for each $m \geq 1$ and intersecting the resulting probability-one events over $m$ shows that almost surely $Q_{n} \to T$ .

step 2.1F3F4

4.1

The degeneracies are covered: $T > 0$ is required, so $h > 0$ and the sums are nonempty with $2^{n} \geq 2$ terms; the partition sequence is the one named in [F5], with mesh $T / 2^{n} \to 0$ and with consecutive refinements $π_{n} \subseteq π_{n + 1}$ , so no ambiguity of convention arises at $t = T$ , where the step and partial-increment conventions coincide by Quadratic variation along a partition sequence; and AC enters only through [F6].

step 2.1step 3.1F5F6given∎

Source notes

Lawler, Theorems 2.8.1 and 2.8.2, proves the mean-square convergence of the dyadic quadratic sums and their almost-sure convergence along meshes whose sizes are summable (here $T / 2^{n}$ ). The computation above is the direct one: the second and fourth Gaussian moments of the increments give $Var (Q_{n} - T) = 2 T^{2} / 2^{n}$ , Chebyshev gives summable error probabilities, and Borel-Cantelli upgrades to almost-sure convergence.

Depends on

Used by

The Brownian p-variation threshold Example

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Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Theorems 2.8.1-2.8.2 (standard reference, not scraped)