Understanding total, quadratic, and $\Phi$ variation of functions

Question

I've started to study stochastic calculus on my own recently (I'll read Fima's book for a simpler introduction and then Steele's for a maybe more formal approach). I've come across the definition of the total variation, quadratic variation and other such as $\Phi$-variation. For the more general definition, the $\Phi$-variation of a function $g$ in the interval $(0,t]$ is calculated as

$$ V_{\Phi}(g) := \sup \sum_{i=1}^{n} \Phi\left(|g(t^n_i) - g(t^n_{i-1})|\right) $$ for a partition $0= t_0 < t_1 < \ldots < t_n = t$ of the interval $(0,t]$. The most used for stochastic calculus as I see for now is the quadratic varaition, where $\Phi(u) = u^2$ and the total variation where $\Phi(u) = u$. I have a few questions about it though.

First, I'd like to see if my understading is right. As I see, the total variation of a function measures the oscillation of the function in the interval, thus $V_{\Phi}(g)(t) < \infty$ for $\Phi (u) = u$ means that the function is "well behaved". For instance, a function such as $f(t)=t\sin(1/t)$ has infinite variation for whatever interval that contains the zero because near the $0$ the function oscillates a lot. Is my understanding correct or am I missing something for the meaning of the total variation?

Second, I guess so far that the use of quadratic variation is important because of the moments of stochastic process: I imagine that in the future, when we want to write Ito's integral, we will want the process to have finite second moment and there might be a link between this and the quadratic variation. But I can't see any use for $\Phi$-variation. Can someone point me out a usage for that in stochastic calculus?

Answer 1

Total variation

Suppose $g(t)$ represents the position of a particle at time $t$. Then the total variation $V_{|\cdot |}(g)(t)$ measures the total distance travelled by the particle on $[0,t]$. If this particle were driving a car, then $V_{|\cdot |}(g)(t)$ would be the measurement tracked by the odometer. It is instructive to consider the case where $g \in C^1$, as here the total distance travelled is the integral of the speed (not velocity!) of the particle: $$V_{|\cdot |}(g)(t) = \int_0^t |g^\prime (s)|ds$$ One of my real analysis professors, a hiker, liked to describe the total variation function as a measurement of "how hard it would be to traverse a hilly trail".

Indeed, being of bounded variation is a form of "niceness" or regularity. Of note, the following are equivalent:

1. $g$ is of bounded variation
2. $g$ is the difference of two bounded increasing functions
3. $g$ is associated with a finite signed measure $\mu_g$, i.e. $g(t) = \mu_g [0,t]$

(cf. Bounded variation, difference of two increasing functions and consider the Jordan decomposition theorem for 3). I'm not sure if I would use the term "oscillation" when thinking of total variation, as this may be confused with the oscillation of a function.

Quadratic- and $p$-variation

For $\Phi (u) = u^p$ for $p > 1$, the $\Phi$-variation becomes a measure of "roughness" or irregularity. Notice that if $g$ admits non-trivial (bounded) $p$-variation, then $p$ is a critical index (i.e. consider what happens for $q > p$ and $q < p$).

For Itô calculus, $p=2$ is particularly important, as you will later learn that all non-constant continuous local martingales admit non-trivial quadratic variation (usually denoted $\langle M \rangle_t$). This plays a role deriving the chain rule (called Itô's lemma) for stochastic calculus. That is, if $g \in C^2$ and $M_t$ is a local martingale (e.g. Brownian motion), the stochastic chain rule is:

$$ \begin{align*} g(M_t) - g(M_0) &= \underbrace{\int_0^t g^\prime (M_s) dM_s}_{FTC} + \underbrace{\frac{1}{2} \int_0^t g^{\prime \prime} (M_s) d\langle M \rangle_s}_{QV} \end{align*}$$

Where the first integral is the Itô integral, resembling the form of the fundamental theorem of calculus, and the second integral is a quadratic variation correction.

When $p \neq 2$, deriving Itô's formula for a process can be tricky. You may wish to look up the stochastic calculus of fractional Brownian motion or the theory of rough paths for more information.