$$ \newcommand{\F}{\mathcal{F}} \newcommand{\I}{\mathcal{I}} $$

Recently I have discovered the awesome world of stochastic processes. Firstly, “stochastic process” is a horrible name, it does not have anything to do with “process.” If you have never heard of this term, brace yourself because this is gonna sound insanely familiar. So, a stochastic process is an indexed list of random variables. That’s it. If you have ever worked with randomized algorithms, this is what you call an algorithm, it takes an input and the output is modeled by a random variable.

I am going to give a slightly more formal definition now that we know what it means.

Definition. Fix a probability space $(\Omega, \F, P)$ and a measurable space $(S, \Sigma)$. A stochastic process is a collection \[ \{X(i) : i \in \I\} \] of $S$-valued random variables indexed by a set $\I$.

In many applications, the index set $\I$ is the positive real numbers and represents time. More generally, it is common to assume that $\I$ is ordered. This adds a lot of structure and allows one to talk about increments (how much $X(i)$ differs from $X(i+j)$) and stuff like that.

Also, if you have done some advanced probability, you can observe that stochastic processes generalize Markov chains, random walks, and martingales.

I’m going to end this short post by answering a burning question: how can you use stochastic processes to prove theorems? By leveraging stochastic calculus.

Further Reading