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.

Read more about this topic:  Quadratic Variation

Famous quotes containing the words finite and/or processes:

    “God is a being of transcendent and unlimited perfections: his nature therefore is incomprehensible to finite spirits.”
    —George Berkeley (1685–1753)

    “The higher processes are all processes of simplification. The novelist must learn to write, and then he must unlearn it; just as the modern painter learns to draw, and then learns when utterly to disregard his accomplishment, when to subordinate it to a higher and truer effect.”
    —Willa Cather (1873–1947)