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
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 {πi} such that ∑ i = 0∞mesh(πi) < ∞, for example πi = {a, a + t / 2i, a + 2t / 2i, …, a + (2i − 1)t / 2i, a + t}, we can apply Borel-Cantelli:
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.
Pingback: Girsanov’s Theorem | Maths, Stats & Functional Programming