stationary process

Here’s what the Wikipedia page says when defining a stationary process:.
It’s a stochastic(big word) process whose joint probability distribution(big words), do not change when shifted in time or space.

I think the key big words that would throw off a regular user is stochastic and joint probability distribution. There’s also a couple of implicit assumptions like:
1.joint probability distribution of variables/parameters of the process
2. These variables/parameters are random

Stochastic process: well this just means the process is assumed to be a collection of random variables.

Joint probability distribution: this just means an n-dimensional space/surface/volume which encompasses the probability values that could be taken by the n number of random variables.

Ok, ok, that’s all too much symbolic. what does it mean in correspondence to anything in real life?

Well, this is all math models that are used for various real-life phenomena. In this case, let’s assume the stock market prices. So what does it mean to say, it is a stochastic process? and what exactly is the joint probability distribution and what the hell does a stationary process mean?

I’ll try to answer these as accurately as i think i can below, but keep in mind, am just an amateur who does this for fun. Don’t blame me for applying my ideas and losing money.

Well to begin with stochastic process, means we are assuming that the stock price generation is a random process. Note that it doesn’t translate to we don’t know what generates it or how it changes, but to we can’t find any pattern that can be used repeatedly and works on the stock prices fluctuations, within reasonable error values that is.

The difference is subtle and easy to miss. We are making a model and the goal of it is to be able to more accurately describe the data we have seen. One useful application is predictive analytics, but with stock market as NNT has argued so loudly, we need to be very careful of fractal/viral effects and also “black swan effects”

Anyway, to get back to our original question, how do we know if the stock market prices are a stationary process or not. Well, the answer turns out to be rather simple(at least in theory, I’ll get to what i know about the practice in a minute). We figure out or pick a set of variables that constitute stock market prices. Note, the actual trading price is dynamically, changing, and we usually approximate (either the closing price, or average over a length of time). But the point is we need find a set of variables to make a joint probability distribution out of and transform or track them. There are quite a few available and some of the immediate ones many will recognize are (P/E ratio, Dividends history,transaction volume, earnings per quarter etc..)

The picking of the number of variables and which variables, depend on what kind of trade-offs you want to make for your model. Typical challenging factors are:
1.Computational capacity(cpu,memory, time etc..)
2.Data availability
3.Uncertainty/entropy the variable adds/subtracts from the model

Once you have these you plot/fit them to a function of n-dimensional space* and call it a joint-probability distribution.

Now comes the testing for stationarity part, you translate this function in one of the dimensions and see if the overall properties of the functions changes. (i.e: properties like volume/area/length, convexity/concavity, topological properties like connectedness, fractal dimensions etc.)**

The Wikipedia article talks about shifting in time, but am ignoring that assuming it can be considered as one of these n-dimensions. The article perhaps uses it because stationary process it seems is a key terminology in time-series analysis.

Ok, so we know whether our model of the stock price of a particular company is stationary or not. So what does this mean to me in my investing behaviour. How should i let it affect or( should i not ) let it affect my investing behaviour?

Now that’s what quite a lot of investment analysts get paid for and as a few of them have spoken out, it’s useful to use some skeptic questioning about their recommendations. Mine on the other hand should be either dismissed altogether or thought about, re-read, take notes of your thoughts, re-read altogether etc.

Any case, my first instinct is to say, the outcome in this case the stock price as modeled by your chosen variables, has very low volatile. Actually with the assumption i have been running with it should have zero volatility, i.e: it’s price stays the same. But then stationarity is not a binary variable as i implied implicitly somewhere in the text above. It is a degree and can be measured based on how much the function mismatches with the original with respect to the amount of translation.
Am guessing it would be some form of partial derivative terms, but don’t have the energy or time to look up, understand and write. Besides am already close to the 1000 word mark and will sign-off now.

With these conclusions:
1. You can check for stationarity of a function, and make predictions based on that about future variations with respect to the parameters involved.
2. You can model a stock price of any company, with respect to some parameters you picked and predicted future prices with respect to changes in the parameters. (It can also be used as a way to test the model accuracy, backtesting it is called i think)
3. Stationary process is an interesting concept or standard in terms of measuring accuracy of your simulation models.

* — I suspect in practice it comes down to 3 or 4 at most but i am just conjecturing there.
** — most likely in this case only the area/volume/length properties.