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 π_n = {a = t₀ < t1 < … < t_n − 1 < t_n = a + t} be a partition of [a, a + t]. Define mesh(π_n) = _defnmax_{0 ≤ i ≤ n − 1}(t_i + 1 − t_i)

Let π_n be a sequence of partitions of [a, a + t] such that ∑ _nmesh(π_n) < ∞ and let π_nW = ∑ _{t ∈ πn}(W_ti + 1 − W_ti)². Then lim_{n → ∞}π_nW = t almost surely.

We have
∑ _{t ∈ πn}(W_ti + 1 − W_ti)² − t = ∑ _{t ∈ πn}(W_ti + 1 − W_ti)² − (t_i + 1 − t_i) = ∑ _i = 0^n − 1X_i
where
X_i = (B_ti + 1 − B_ti)² − (t_i + 1 − t_i)
Therefore since the X_i are independent with zero means

We know that the 4th moment of normally distributed random variable with variance σ is 3σ² so

Thus

And we can therefore conclude that

Now apply Chebyshev’s inequality:

By choosing the sequence {π_i} such that ∑ _i = 0^∞mesh(π_i) < ∞, for example π_i = {a, a + t / 2ⁱ, a + 2t / 2ⁱ, …, a + (2ⁱ − 1)t / 2ⁱ, a + t}, we can apply Borel-Cantelli:

to conclude that |∑ _{t ∈ πn}(W_ti + 1 − W_ti)² − (t_i + 1 − t_i)| < ε eventually i.e. for any ε there exists an n such that for all m ≥ n, |∑ _{t ∈ πm}(W_ti + 1 − W_ti)² − t| < ε and therefore that π_nW → 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.