Definition
The variation of a function {% g %} is defined as
{% V(T) = sup_{partitions} \sum | g(t_i) - g(t_{i-1}) | %}
where the sup is taken over all partitions of {% [0,T] %}.
(see turner pg 182)
Quadratic Variation
The quadratic variation of a function {% g %} is defined as a limit as the partition size of the domain (in this case, the closed interval {% [a,b] %}) goes to zero, of the sum of squares of the difference beteen consecutive points.
{% Q([a,b]) = lim_{|partiion| \rightarrow 0} \sum | g(t_i) - g(t_{i-1}) |^2 %}
(see turner pg 184)
Here, we have {% a \leq t_i \leq b %}
Co-Variation (Cross Variation)
the Cross Variation is defined to be
{% [f,g]([a,b]) = lim \sum_{i=1}^n (f(t_i) - f(t_{i-1}))(g(t_i) - g(t_{i-1})) %}
The quadratic variation is then seen as the case where {% g %} is taken to be the same function as {% f %}
in the cross variation.
{% Q(T) = [f,f]([0,T]) %}
It is then often written as
{% Q(T) = [f]([0,T]) %}
Co-Variation Process
If the left side of the interval used in the definition of the cross variation is fixed, say at {% 0 %}, then the cross variation can be defined to be a function.
{% [f,g]([0, t]) = [f,g](t) %}
If the function {% [f, g] %} is differentiable, you will often see it expressed in terms of the
Riemann Stieltjes Integral
{% [f,g](T) = \int_0^T d [f,g] %}
Heuristics
Given the definitions above, a set of heuristics and and easily memorized formulas are often used. It should be noted that these arent formally mathematically defined, and are in essence, only stand-ins for the formal defitions.
The cross variation is often written out in differential form as
{% d [f,g] = df dg %}
Typically, this is mostly used in the context of
stochastic calculus,
where the cross variation of two
Ito Processes
will be written as
{% dW_i dW_2 = \rho dt %}
The quadratic variation is then written as
{% d [f] = dW dW = [W](t)dt %}
When {% W %} is a
Brownian Motion
this becomes
{% dW dW = dt %}