Quadratic Variation

def. Total Variation of a function $f$ in interval $(a,b)$ is defined:

$$V_a^b(f):=\underset{|P|\to 0}{\lim }\sum _{k=0}^{n-1}\mid f(x_{k+1})-f(x_k)\mid$$

where $P$ is the mesh size. Formally, the maximum mesh size. $P\to 0$ means we are chopping up more finely. Visualization. Consider $\sin (x)$ in interval $(0,2\pi )$.

def. Quadratic Variation of stochastic process $\{X_t{\}}_{t\ge 0}$ (on a probability space) in interval $(0,t)$ is:

$$[X{]}_t=\underset{P\to 0}{\lim }\sum _{k=1}^n(X_{t_k}-X_{t_{k-1}}{)}^2$$

where $P$ is also the mesh size as in total variation.

• Intuition. $[X{]}_t$ is thought of as a measure of how much energy or ADHD a function has in interval $(0,t)$
• Quadratic variation is also a function itself indexed by $t$.