Quadratic Variation - Finite Variation Processes

Finite Variation Processes

A process X is said to have finite variation if it has bounded variation over every finite time interval (with probability 1). Such processes are very common including, in particular, all continuously differentiable functions. The quadratic variation exists for all continuous finite variation processes, and is zero.

This statement can be generalized to non-continuous processes. Any càdlàg finite variation process X has quadratic variation equal to the sum of the squares of the jumps of X. To state this more precisely, the left limit of Xt with respect to t is denoted by Xt-, and the jump of X at time t can be written as ΔXt = Xt - Xt-. Then, the quadratic variation is given by

The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here, P is a partition of the interval, and Vt(X) is the variation of X over .

\begin{align} \sum_{k=1}^n(X_{t_k}-X_{t_{k-1}})^2&\le\max_{k\le n}|X_{t_k}-X_{t_{k-1}}|\sum_{k=1}^n|X_{t_k}-X_{t_{k-1}}|\\ &\le\max_{|u-v|\le\Vert P\Vert}|X_u-X_v|V_t(X). \end{align}

By the continuity of X, this vanishes in the limit as goes to zero.

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