Understanding Quadratic Variation

Question

I think part of the trouble a lot of people (or at least me personally) have with making the jump from calculus to stochastic calculus is the notion of quadratic variation. It doesn't have as much visual, or intuitive, appeal as "the area under a graph" or the "instantaneous rate of change".

Anyone come across a really good way to think about what quadratic variation is and what it measures?

I've heard that quadratic variation of a process over $[0,T]$ quantifies the accumulated randomness of the process, but even that is a bit difficult to unpack.

It's pretty easy to prove that a process which is continuously differentiable over an interval has zero quadratic variation over that interval. In the case of a stochastic process, if the process is cont. differentiable with respect to time over an interval $[0,T]$, doesn't this sort of imply a sense of "non-randomness" (probably bad choice of words)?

In other words if we have a stochastic process $M(t)$, and if we know that $lim_{s\rightarrow t^+} \frac{M(s) - M(t)}{s-t} = c<\infty$ (limit from the left), then if we're standing at $s<t$, we know how much $M(s)$ will move for an infinitesimally small $\Delta t$. In this sense, we can "predict" how $M(s)$ will move in the extremely short future based on the info we have at time $s$. Hence the "nonrandomness" interpretation. So a random process which has continuously differentiable sample paths over $[0,T]$ can be "predicted" for tiny, tiny future increments at each point $t\in [0,T]$ and accumulates no "randomness".

I'm sure this is wrong on many levels, but any insight or comments would be appreciated.

Answer 1

I believe your argument provides a good interpretation for total (not quadratic) variation, because it is the one associated to the first derivative.

For quadratic variation, one should note first that squaring an infinitesimal element will cause it to shrink even more. So if you are adding up smaller things than before and it still results in a great number, then you can infer that the original increments were not that small. It might be an indication that a trajectory is very chaotic (a lot of randomness accumulated). In other words: the higher the quadratic variation, the higher is the amount of randomness.

Conversely, if a trajectory is "minimally smooth", although not differentiable, then the squaring procedure applied to the infinitesimal increments would be enough to keep the resulting summation controlled (sufficiently small). So you could also think about quadratic variation as a measured of "smoothness" of a trajectory.

Answer 2

Quadratic variation is the reason why you can't approximate a "jumpy" function as a 1st order taylor expansion, no matter how small you make the interval. It won't happen. Because the function "jumpiness" is always of higher "order" than your small interval. That is, it looks jumpy no matter at which micro/macroscopic scale you try to look at it. QV measures this jumpiness. In contrast, you can refine your scale for a well behaved function so as to make it small enough that the function looks absolutely smooth in that small enough interval, thus allowing you to approximate the increment with a first order expansion.

The practical consequence is that it is inadequate to describe the increment in this jumpy (say convex for arguement's sake) function in any arbitrarily small interval by just the first order expansion. The jumps (say for instance, 1 up and 1 down in the chosen interval) and convexity mix together, as the convexity places a higher value on the up jump rather than the down jump. These jump-convexity effects accumulate systematically from micro to macroscopic level. Note that the same essentially happens even for well behaved functions; but in those cases the small interval chosen squeezes this effect to a negligible amount. But in the jumpy case, the small interval cannot squeeze it - remember, the jumpiness is of higher order than the interval length.

This $jumpiness*convexity$ is exactly the extra drift term you get in Ito's lemma. Remember how we said that QV measures jumpiness. The extra term is simply $1/2*convexity*QV$.

In short, quadratic variation is the measure of "noisiness/jumpiness" of the function with respect to any arbitrarily chosen $x$ interval.