What is a Markov Process?

A few students that have great interest in going into financial mathematics, be in JC students or Uni students that were taught by me before, asked me where to start. So I often recommend them to read on Markov Process first. Of course, you need a certain knowledge for statistics here to be able to read. My brother attempted reading on it a bit and faced much discomfort. There were many definitions and terminologies that were not that user friendly.

A Markov Process is a stochastic process with the following properties
(a) The number of possible outcomes or states is finite.
(b) The outcome at any stage depends only on the outcome of the previous stage.
(c) The probabilities are constant over time.

I will discuss more on stochastic process in a later post, and possibly compare it to deterministic process. But for now, lets looks at what the above properties imply.

Credits: WIkipedia

Credits: WIkipedia

So the above diagram shows that we have three (finite) states, they are, bear, bull, and stagnant markets. We also see that the outcome of any stage here is pre-determined by the previous stage, be it moving from a stagnant market to a bear market. Lastly, the probabilities are constant OVER TIME. These probabilities are what we call transition probabilities, which represents the likelihood of moving from one state to another state.

Markov Processes help us to understand the happenings of the world, given the assumptions are adhered to. At the same time, it means that what happens in the long run won’t depend on where the process started, or on what happened along the way. It will, however be completely determined by the transition probabilities.

Does this imply that history doesn’t matter and I do not look at a company’s past performance before making a decision? Not quite, since the assumption clearly will not hold in reality. Markov processes model provides a framework to which we can compare the world. And if the world fits the assumption, that what happens in the long run is pretty much determined and affirmed.

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