The building blocks of stochastic calculus — Part I: Wiener processes
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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).
Essential properties of W(t)
Also known as a Brownian motion, the Wiener process W(t) is a stochastic process in time with the following statistical properties:
where E[…] denotes the mean or expected value, and Var[…] is the variance. Essentially, the process has mean 0 across all time t, but it has variance which increases linearly with time, that is: the longer the time it goes on for, the larger the variance is. We can think of this as a particle which is subjected to random forces. The more time goes by, the harder it becomes to predict its position. If we consider for instance a one-dimensional trajectory (left or right motion only), and the particle’s displacement denoted as X(t), then the displacement X(t) when the particle is subjected to a Wiener process (e.g. random nudges at each point in time) will have sample trajectories such as the ones shown in Figure 1.
