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* = *t*_{0} < *t*1 < … < *t*_{n − 1} < *t*_{n} = *a* + *t*} be a partition of [*a*, *a* + *t*]. Define mesh(*π*_{n}) = _{defn}max_{0 ≤ i ≤ n − 1}(*t*_{i + 1} − *t*_{i})

Let *π*_{n} be a sequence of partitions of [*a*, *a* + *t*] such that ∑ _{n}mesh(*π*_{n}) < ∞ and let *π*_{n}*W* = ∑ _{t ∈ πn}(*W*_{ti + 1} − *W*_{ti})^{2}. Then lim_{n → ∞}*π*_{n}*W* = *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 − 1}*X*_{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 3*σ*^{2} 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* / 2^{i}, *a* + 2*t* / 2^{i}, …, *a* + (2^{i} − 1)*t* / 2^{i}, *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*_{n}*W* → *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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