p-Variation
1 Definition
The \(p\)-th variation of a stochastic process measures the accumulated size of its increments raised to the power \(p\) along finer and finer partitions of time.
Let \((X_{t})\) be a real-valued continuous-time process and \(p>0\). For \(t>0\) consider partitions \[ \Delta^{(n)}:=\{ 0 = t_{0}^{(n)}<t_{1}^{(n)}<\dots<t_{K_{n}}^{(n)}=t \} \] of \([0,t]\) with mesh \[ \delta^{(n)}:=\max_{1\leq i\leq K_{n}}\lvert t_{i}^{(n)}-t_{i-1}^{(n)} \rvert \to 0\quad \text{as}\quad n \to \infty. \] The \(p\)-variation \(\left\langle X,X \right\rangle_{t}^{(p)}\) is the process (if it exists, and does not depend on the choice of partitions) such that \[ S_{n}^\Delta = \sum_{i=1}^{K_{n}} \lvert X_{t_{i}^{(n)}}-X_{t_{i-1}^{(n)}} \rvert ^p \stackrel{L_{2}~\&~\mathbb{P}}\to \left\langle X,X \right\rangle_{t}^{(p)}. \]
In particular:
- for \(p=1\) it is the total variation; and
- for \(p=2\) it is the quadratic variation, often written \(\left< X,X \right>_{t}\).
Quadratic variation is the relevant notion in the \(L_{2}\) setting and is key to the construction of the Itô Integral in Stochastic Calculus. A smooth (differentiable) path has finite total variation and zero quadratic variation, since \(\sum_i|\Delta X_i|^2\leq\max_i|\Delta X_i|\sum_i|\Delta X_i|\to0\).
2 \(p\)-Variation of Brownian Motion
For Brownian Motion \(\left< B,B \right>_{t}=t\). That is, for partitions \(\Delta^{(n)}\) of \([a,b]\) with \(\delta^{(n)}\to 0\), \(S_{n}^\Delta \stackrel{L_{2}~\&~\mathbb{P}}\to b-a\).
Proof. Write \(\Delta_i=t_{i}^{(n)}-t_{i-1}^{(n)}\) and \(\xi_i=B_{t_{i}^{(n)}}-B_{t_{i-1}^{(n)}}\sim\mathcal{N}(0,\Delta_i)\), which are independent. Then \(\mathbb{E}S_n^\Delta=\sum_i\mathbb{E}\xi_i^2=\sum_i\Delta_i=b-a\) and, using \(\mathbb{E}\xi_i^4=3\Delta_i^2\), \[ \operatorname{Var}(S_n^\Delta)=\sum_i\operatorname{Var}(\xi_i^2)=\sum_i(3\Delta_i^2-\Delta_i^2)=2\sum_i\Delta_i^2\leq2\delta^{(n)}(b-a)\to0. \] Hence \(\mathbb{E}|S_n^\Delta-(b-a)|^2=\operatorname{Var}(S_n^\Delta)\to0\), which is convergence in \(L_2\), and so in probability (see Convergence in Lp Norm).
Almost surely, Brownian motion has infinite total variation on every interval \([a,b]\) with \(a<b\), and hence is not differentiable there.
Proof. Pass to a subsequence of partitions along which \(S_n^\Delta\to b-a>0\) almost surely. Since Brownian paths are continuous, hence uniformly continuous on \([a,b]\), \(\max_i|\xi_i|\to0\). Then \[ S_n^\Delta=\sum_i\xi_i^2\leq\max_i|\xi_i|\sum_i|\xi_i|, \] so \(\sum_i|\xi_i|\geq S_n^\Delta/\max_i|\xi_i|\to\infty\).
Heuristically, with \(T=n\Delta t\), \(\sum_i|\Delta B_i|\sim n\,\mathbb{E}|B_{\Delta t}|\sim\frac{T}{\Delta t}\sqrt{\Delta t}\to\infty\).