The Quadratic Variation of Brownian Motion

Protter(Protter 2004) notes that if some assumptions are made on how fast the mesh (see below for a definition) converges then the calculation of the almost sure value of the quadratic variation of Brownian Motion can be done using Chebyshev’s Inequality and the Borel-Cantelli Lemma. Williams and Rogers(Rogers and Williams 2000) also set this as an exercise.

Let πn = {a = t0 < t1 < … < tn − 1 < tn = a + t} be a partition of [a, a + t]. Define mesh(πn) = defnmax0 ≤ i ≤ n − 1(ti + 1 − ti)

Let πn be a sequence of partitions of [a, a + t] such that ∑ nmesh(πn) < ∞ and let πnW = ∑ t ∈ πn(Wti + 1 − Wti)2. Then limn → ∞πnW = t almost surely.

We have
∑ t ∈ πn(Wti + 1 − Wti)2 − t = ∑ t ∈ πn(Wti + 1 − Wti)2 − (ti + 1 − ti) = ∑ i = 0n − 1Xi
where
Xi = (Bti + 1 − Bti)2 − (ti + 1 − ti)
Therefore since the Xi are independent with zero means
\mathbb{E}\bigg(\sum_{t \in \pi_n}(W_{t_{i+1}} - W_{t_i})^2 - t\bigg)^2 = \mathbb{E} \bigg(\sum_{i=0}^{n-1} X_i\bigg)^2 = \sum_{i=0}^{n-1}\mathbb{E} X_i^2
We know that the 4th moment of normally distributed random variable with variance σ is 3σ2 so
\begin{aligned} \mathbb{E} X_i^2 &= \mathbb{E} (W_{t_{i+1}} - W_{t_i})^4 - 2(t_{i+1} - t_i)\mathbb{E} (W_{t_{i+1}} - W_{t_i})^2 + (t_{i+1} - t_i)^2 \\ &=3(t_{i+1} -t_i)^2 - 2(t_{i+1}-t_i)^2 + (t_{i+1}-t_i)^2 \\ &= 2(t_{i+1} -t_i)^2\end{aligned}

Thus
\begin{aligned} \sum_{i=0}^{n-1} \mathbb{E} X_i^2 &= 2\sum_i^{n-1}(t_{i+1} -t_i)^2 \\ &\le 2\text {mesh}(\pi_n)\sum_0^{n-1}(t_{i+1} - t_i) \\ &= 2t\text {mesh}(\pi_n)\end{aligned}

And we can therefore conclude that
\mathbb{E} \bigg(\sum_{i=0}^{n-1} X_i\bigg)^2 \le 2t\text {mesh}(\pi_n)

Now apply Chebyshev’s inequality:
\mathbb{P} \bigg(\bigg\lvert \sum_{i=0}^{n-1}X_i\bigg\rvert > \epsilon\bigg) \le \frac{1}{\epsilon^2}\mathbb{E} \bigg(\sum_{i=0}^{n-1}X_i\bigg)^2 \le \frac{2t\text {mesh}(\pi_n)}{\epsilon^2}

By choosing the sequence {πi} such that ∑ i = 0mesh(πi) < ∞, for example πi = {a, a + t / 2i, a + 2t / 2i, …, a + (2i − 1)t / 2i, a + t}, we can apply Borel-Cantelli:
\sum_{n=1}^\infty \mathbb{P} \bigg(\bigg\lvert \sum_{t \in \pi_n}(W_{t_{i+1}} - W_{t_i})^2 - (t_{i+1} - t_i)\bigg\rvert > \epsilon\bigg) \le \frac{2t}{\epsilon^2}\sum_{i=1}^\infty\text {mesh}(\pi) < \infty

to conclude that |∑ t ∈ πn(Wti + 1 − Wti)2 − (ti + 1 − ti)| < ε eventually i.e. for any ε there exists an n such that for all m ≥ n, |∑ t ∈ πm(Wti + 1 − Wti)2 − t| < ε and therefore that πnW → t as n → ∞.

Protter, P. E. 2004. Stochastic integration and differential equations. Applications of mathematics. Springer. http://books.google.co.uk/books?id=mJkFuqwr5xgC.

Rogers, L. C. G., and D. Williams. 2000. Diffusions, Markov Processes, and Martingales. Cambridge Mathematical Library. Cambridge University Press. http://books.google.co.uk/books?id=bDQy-zoHWfcC.

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One thought on “The Quadratic Variation of Brownian Motion

  1. Pingback: Girsanov’s Theorem | Maths, Stats & Functional Programming

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