How to study stochastic calculus
And why you are probably doing it wrong.
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…
