Why "modeling volatility" is not an oxymoron? Firstly, I'm sorry, if my question will come across as simple or even naive, but I have no formal background in statistics and I'm trying my best to learn it as much as I can, among other areas.
My question is two-fold and I will start with its simpler part:

What phenomenon or principle is the foundation that allows modeling a purely random (stochastic) process, described as Brownian
  motion (not
  drift-containing and trend-exhibiting Geometric Brownian
  motion)?

Based on the question above, I'd like to state another, more general question, as stated in the title (please note that, despite including econometrics tag, my questions are concerned with imaginary purely stochastic processes component of real-world complex hybrid economic, social, biological and other natural and artificial processes, containing "modellable" non-stochastic component):

Why "modeling volatility" is not an oxymoron?

I expect an answer to this question to expand a bit on the topic. Recommendations of and references to (preferably not math-heavy) introductory materials are welcome and are appreciated.
 A: The name Brownian motion comes from the guy named Brown, and his microscope observation of spores  moving in the liquid. Einstein got Nobel prize for the theory. The physics are simple, as Einstein put it:

Soon after the appearance of my paper (*) on the movements of
  particles suspended in liquids demanded by the molecular theory of
  heat, Siedentopf (of Jena) informed me that he and other physicists-in
  the first instance, Prof. Gouy (of Lyons)-had been convinced by direct
  observation that the so-called Brownian motion is caused by the
  irregular thermal movements of the molecules of the liquid.

This is how Zeldovich and Myshkis set up the simple theoretical framework as follows:

Along the avenue on which the x-axis, there is a lot of stores 
  equally spaced at $h$ distance from one another. From one of them serving
  as the reference point, the lady runs out: she only just heard that
  somewhere there's a sale on French umbrellas. She randomly chooses the direction along
  the alley, in a time $\tau$ she runs to a nearby store. Hearing that there
  are no French umbrellas, she forgets where she came from, and
  runs again at random to one of two sides. After the time $\tau$ she
  again gets a negative response and, once again losing the sense of direction,
  randomly selects the direction and so on. Where will she end up at time $n\tau$?
  (For simplicity, we assume that the rumor was
  false, so the search for French umbrellas goes on infinitely.

Then next passage sets the expectations of what we can get from the model:

Of course, it is impossible to predict exactly, one can
  only determine the probability of the fact that the lady at the time $n\tau$
  will have a certain coordinate.

Note, how they emphasized the fact the Lady was a bit out of her mind and sense of direction: she doesn't remember where she came from. That's the crucial detail: each move is independent of each other. Then the central limit theorem (CLT) kicks in, so you have the $x(n\tau)=\sum_{i=1}^nx_i=\sum_{i=1}^n(-1)^\xi_ih$, where $\xi_i$ - bernulli process with $p=1/2$. According to CLT this must converge to some kind of normal distrobution.
What's the problem with "modeling volatility"? It is a strong assumption to state that the volatility is constant. In economics and finance it fails again and again. Think of volatility as a noise "level", it doesn't have to remain constant and it doesn't. What guys like Heston noticed, was that volatility seems to be stationary and persistent, hence the family of stochastic volatility models. 
A: Many financial instruments are often modeled as being stochastic partial differential equations or difference equations if modeled in discrete time.  
It is not so different as in time series analysis where series are modeled as some sort of low order process having trends, seasonal components and irregular noise superimposed in the same process.  
In more classical setting volatility is an parameter, not an variable to be modeled but this has changed too. It might not be theoretically justifiable to incorporate deterministic parameters and their sample estimates in used model, but of course in reality many might do that. For example in Black & Scholes valuation formula has fixed parameters and users often supplant sample estimates into the formulae.
Users generally can fall into the trap of model risk if they believe too much that their formulas incorporate all relevant aspects of reality.
A: I think you are having problems with the notion that a random process can be described by rules - eg in a model. 
Standard Brownian motion ${B_t}$ is defined as the process that 


*

*starts at 0 at time zero, 

*the time differences  $B_t-B_s$ are normally distributed with mean 0 and variance $t-s$ independently of  past ${B_u}_{0\le u<s}$  (for $t>s>0$)

*has continuous sample paths


So it should be clear that the Gaussian independent increment assumption is a clear modelling assumption - one could make other assumptions and have different sorts of randomness with different statistical properties
wiki brownian motion
