Brownian Motion

Quadratic Variation of Brownian motion- Part-1

We have already computed the quadratic variation of the scaled random walk \(W^{(n)}\) up to time T; and it is- \[\begin{aligned} {[W^{(n)},W^{(n)}]}(T) &= \sum_{j=1}^{nT}[W^{(n)}(\frac{j}{n})-W^{(n)}(\frac{j-1}{n})]^2 \\ &= T \end{aligned}\]

In the formula above the step sizes are all equal in scaled symmetric random walk; but the Brownian motion there is no natural step size (i.e. equal step size). For given, T > 0, we could simply choose step size \(\frac{T}{n}\) for some \(\lim \text{n} \rightarrow \infty\) compute the quadratic variation of Brownian motion up to time T by-

\[\begin{aligned} {[W,W]}(T) &= \lim_{n \rightarrow \infty}\sum_{j=1}^{nT}[W(\frac{j}{n})-W(\frac{j-1}{n})]^2 \\ \end{aligned}\] The paths of Brownian motion are unusual in that their quadratic variation is not zero. This makes stochastic calculus different from ordinary calculus. Let’s see what is this- n-th order variation, starting with First order variation.

First Order Variation

Consider the following picture. The function f(t), which is plotted over \(t \in [0,T]\). We want to calculate the First order variation. First we make a partition of [0,T]. Let the partition be- \(\Pi = \{0=t_0,t_1,t_2,T\}\)

The First Order Variation of the function f(t) over the partition \(\Pi\) is calculated as- \[\begin{aligned} FV_T(f) &= [f(t_1)-f(t_0)]-[f(t_2)-f(t_1)]+[f(T)-f(t_2)] \\ &= \int_{0}^{t_1} f'(t) \, dt +\int_{t_1}^{t_2} (-f'(t)) \, dt + \int_{t_2}^{T} f'(t) \, dt \\ &= \int_{0}^{T} |f'(t)| \, dt \end{aligned}\]

Generalizing, we have- suppose we want to calculate the First order variation of a function f(t), \(t \in [0,T]\). First we choose a partition \(\Pi = \{0=t_1,t_2, \cdots ,t_n = T \}\), where- \(0 = t_0 < t_1 < t_2 < \cdots < t_n = T\). These points in the partition \(\Pi\) need not to be equispaced.

But why do we need these partition?

At the starting of this blog I stated that- there is no natural step size for Brownian motion like scaled symmetric random walk. But if you want to integrate some function- you need some partition like we do in Riemann sum (https://en.wikipedia.org/wiki/Riemann_sum). This partitions will help us to determine step-size equivalent to scaled symmetric random walk.

The maximum step-size of a partition is defined as- \[||\Pi|| = \max_{j= 0,\cdots, n-1} (t_{j+1}-t_j)\] Then the First Order Variation of function \(f(t)\) over interval \([0,T]\) is defined as- \[\begin{aligned} FV_T(f) = \lim_{||\Pi|| \rightarrow 0} \sum_{j=0}^{n-1} |f(t_{j+1})-f(t_j)| \end{aligned}\] the limit in the above summation is taken as the maximum step-size of partition tends to zero, n, the number of partition points tends to infinity , i.e. \(\lim_{||\Pi|| \rightarrow 0} \implies \lim_{n \rightarrow \infty}\). Not the other way around.

Her we have written two forms of \(FV_T(f)\), one in integration form another in summation form. To prove that- we just need to show two forms are equivalent.

For this we are going to use the Mean Value Theorem, which applies to any function \(f(t)\) whose derivative \(f'(t)\) is defined everywhere. Mean Value Theorem states that- for any interval [a,b] with- a < b, there exists a point \(c \in [a,b]\) such that- \[\frac{f(b)-f(a)}{b-a} = f'(c)\] This is also true for any interval \([t_j,t_{j+1}]\) in the partition \(\Pi\). In other words, somewhere between \(t_j\) and \(t_{j+1}\) the tangent line is parallel to the chord connecting \((t_j,f(t_j))\) and \((t_{j+1},f(t_{j+1}))\), like the figure below-

So for any partition in \(\Pi\), for some \(t_j^{*} \in [t_j,t_{j+1}]\), we have- \[\begin{aligned} &\frac{f(t_{j+1})-f(t_j)}{t_{j+1}-t_j} = f'(t_j^{*}) \\ \implies &f(t_{j+1})-f(t_j) = f'(t_j^{*})(t_{j+1}-t_j) \end{aligned}\]

Thus we have- \(\sum_{j=0}^{n-1} |f(t_{j+1})-f(t_j)| = \sum_{j=0}^{n-1} |f'(t_j^{*})|.(t_{j+1}-t_j)\).

And we have previously seen that- \(lim_{n \rightarrow \infty} \implies \lim_{||\Pi|| \rightarrow 0}\). So we can also write-

\[\begin{aligned} &\lim_{||\Pi|| \rightarrow 0} \sum_{j=0}^{n-1} |f'(t_j^{*})|.(t_{j+1}-t_j) \\ = &\lim_{n \rightarrow \infty} \sum_{j=0}^{n-1} |f'(t_j^{*})|.(t_{j+1}-t_j) \\ \end{aligned}\]

This is nothing but Riemann sum for the integral of the function \(|f'(t)|\) over interval [0,T]. Therefore we have-

\[\begin{aligned} FV_T(f) &= \lim_{n \rightarrow \infty} \sum_{j=0}^{n-1} |f'(t_j^{*})|.(t_{j+1}-t_j) \\ &= \int_{0}^{T} |f'(t)| \, dt \qquad \text{[By Riemann sum]} \end{aligned}\]

Similar to First Order Variation we can also calculate Quadratic Variation which we will see in the next blog. Happy reading!!