How to study stochastic calculus

And why you are probably doing it wrong.

A sample simulation of a stochastic process. Image credit: https://people.duke.edu/~rnau/411rand.htm

Stochastic calculus, as its name suggests: is an area of mathematics concerned with the calculus (e.g. derivatives, integrals, limits, etc.) of stochastic processes (also known as random processes, effectively sequences of random numbers which possess certain properties). It is a natural extension of ordinary calculus to functions of a random variable, which are not deterministic over time, but rather random: each value of a function X(t) in time t is in itself a random variable. This means that the ordinary rules of calculus for deterministic functions no longer apply, and this makes things very complicated very quickly.

Stochastic calculus is the bread and butter of many fields of application, particularly quantitative finance and some areas of physics. It has also found many applications in data science as well, because it is a good way to simulate processes which are not known over time, and which can take on many different trajectories depending on how it is influenced by external forces. For instance, stocks and option prices are modelled using tools from stochastic calculus, with a notable example being the Black-Scholes formula. As powerful as it can be for making predictions and building models of things which are in essence “unpredictable”, stochastic calculus is a very difficult subject to study at university, and here are some reasons:

1. Stochastic calculus is not a standard subject in most university departments. Usually, it is only offered in mathematics/statistics majors, or in very specialized degrees like a Master’s of quantitative finance or actuarial science. Very rarely will you find a course suitable for say engineering or physics majors.
2. The way stochastic calculus is taught is very “pure-mathy”. This means that whoever teaches it at university is less concerned with applications and more concerned with mathematical rigour, i.e. “if X(t) is a martingale with respect to some filtration F on a sigma-algebra in …”. This is all fine if you are a pure mathematician, but I can’t say it has much benefit to those who want to apply this to real situations.
3. Most textbooks on the subject are very dull or lacking in examples and insight. In fact, most textbooks on stochastic calculus read more like a jumble of hardcore…