Stochastic Processes

Stochastic Processes - Mental Model Series -13

So much of life, it seems to me, is determined by pure randomness. Sidney Poitier

Photo by Pop & Zebra on Unsplash

Stochastic” means random, so we can call simply a “stochastic process”  as a random process.In the real word, uncertainty is a part of everyday life.Stochastic process helps us to understand them mathematically . As a process,it contains a wide variety of processes within itself in which the movement of an individual variable can be impossible to predict but can be thought through probabilistically only.

Movement of Exchange rates, Stock prices are good examples of stochastic processes. As we know , it’s not possible to predict stock prices on a day-to-day basis, but we can describe the probability of various distributions of their movements over time. Obviously, it is much more likely that the stock market (a stochastic process) will be up or down 1% in a day than up or down 10%, even though we can’t predict what tomorrow will bring.

Some important and simple stochastic processes  which can come handy while thinking in layman terms for you.

Markov Stochastic Process-

Remember :In Markov process,value of a variable does not depend on its historic values.Only the current value of the variable can be taken into account to predict the future values. In other words ,a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history, hence independently from such history(Memory less).

As a result, future predictions are expressed in probability distributions. Also , a variable might follow normal or log normal probability distribution.

It is important to understand that each probability distribution has a mean and variance where the mean is the average value and variance is the dispersion of values from its mean.

Mean is also known as return and variance is known as risk as it adds uncertainty in your variable’s values.

Consider the following example -

Imagine that there are two possible states for weather: sunny or cloudy. You can always directly observe the current weather state, and it is guaranteed to always be one of the two aforementioned states.

Now, you decide you want to be able to predict what the weather will be like tomorrow. Intuitively, you assume that there is an inherent transition in this process, in that the current weather has some bearing on what the next day’s weather will be. So, being the dedicated person that you are, you collect weather data over several years, and calculate that the chance of a sunny day occurring after a cloudy day is 0.25. You also note that, by extension, the chance of a cloudy day occurring after a cloudy day must be 0.75, since there are only two possible states.
You can now use this distribution to predict weather for days to come, based on what the current weather state is at the time.

Random walk -

The random walk theory is the occurrence of an event determined by a series of random movements - in other words, events that cannot be predicted. For example, one might consider a drunken person's path of walking to be a random walk because the person is impaired and his walk would not follow any predictable path.

The random walk theory as applied to stock trading,  the price of securities moves randomly (hence the name of the theory).Therefore, any attempt to predict future price movement, either through fundamental or technical analysis, is futile.So Stock traders is that it is impossible to outperform the overall market average other than by sheer chance. Those who subscribe to the random walk theory recommend using a “buy and hold” strategy.

Poisson process -

The Poisson process is one of the most widely-used counting processes. It is normally  used in scenarios where we are counting the occurrences of certain events that appear to happen at a certain rate, but completely at random (without a certain structure).

For example, suppose we know that earthquakes occur in a certain area with a rate of 2 per month from historical data. Other than this information, the timings of earthquakes seem to be completely random. In this scenario ,e Poisson process might be a good model for earthquakes.

 In practice, the Poisson process or its extensions have been used to model ,
− the number of car accidents at a site or in an area;
− the location of users in a wireless network;
− the requests for individual documents on a web server;
− the outbreak of wars;


At a higher level ,there are two ways to classify a stochastic process:

Discrete: When changes in value of a variable are at fixed points in time. Only certain values can be chosen for a discrete variable.
Continuous: When changes in value of a variable are continuous. Value of a continuous variable can take any value within a certain range.



Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 
For More , read https://jamesclear.com/feynman-mental-models