The building blocks of stochastic calculus — Part I: Wiener processes
Stochastic calculus is the branch of mathematics that focuses on the study of stochastic processes (e.g. processes which are random in time or sequences of random variables obeying certain statistical properties) and the calculus associated with them, namely derivatives and integrals. It is a broad field with many applications in engineering, science and most notably, quantitative finance and actuarial science. It is a difficult subject to learn about because most of the books written on the topic assume the reader is a pure mathematician with a lot of knowledge about abstract mathematics. However, after many years of studying it and using it in my research, I found that you can actually get quite far with understanding only a few fundamental concepts.
In this article, I will describe the most essential building block of stochastic calculus: the Wiener process W(t), and I assume the reader is familiar with elementary statistics and probability, namely concepts such as probability distributions, statistical moments of random variables, and moment generating functions (a topic which I covered in another article here: https://www.cantorsparadise.com/moment-generating-functions-788f16f9d2d6).
