What is a Stochastic Process? It is a collection of Random Variables. What is a Random Variable ? It’s a mapping from sample space to real numbers. What is a Sample Space? It is a set of possible outcomes. How did we arrive at the possible outcomes set? We do a random experiment. What is a random experiment? It’s an experiment whose outcome can’t be computed with certainty. Phew!
Let’s take a real life example to see how we can fit it all. Since the world is getting hotter every summer, people are desperate for a good rain to cool it all down. And since it’s not in your hands, you want to model this whole situation. The act of observing the weather is called a random experiment since it’s not perfectly certain if it’s going to rain or not. The set of possible outcomes here is obviously going to be “Rain” and “No Rain”. Let Xi be a random variable, that equals 1 if it rains and 0 otherwise at day i. Then, the collection of Xi’s over time is gonna form a stochastic process which is defined by {Xi : i = 0,1,2,…}. A stochastic process is defined by two main things. It is characterised by a State Space S and TPM (Transition Probability Matrix). State Space S is again, the set of possible values of Xi can take ( In our case, it’s either 1 or 0). Stochastic Processes can be classified into the following:
- Discrete Index Discrete State Space Processes State Space is a discrete set and the index set is also discrete (You are observing the process in discrete time). The rest of the classification can be understood by common intuition I guess (Behaving like a typical lazy teacher, ain’t I 😝)
- Discrete Index Continuous State Space Processes
- Continuous Index Discrete State Space Processes
- Continuous Index Continuous State Space Processes