The Basics of Stochastic Calculus
Lixin X. Wu · 2019
The seemingly random fluctuation in stock prices is the most distinguishing feature of financial markets and it creates both risks and opportunities for investors. In quantitative modeling, a financial time series is treated as a function of other standardized stochastic processes, and these stochastic processes serve as the engines for the random evolution of financial time series. Among the many standardized stochastic time series that are available, the so-called Brownian motion is no doubt the most basic yet the most important one. Financial time series data actually exist in discrete form, for example, tick data, daily data, weekly data, and so on. The theory of Brownian motion is the single most important building block of continuous-time financial time series. Brownian motion plays a major role in continuous-time stochastic modeling in physics, engineering and finance. In finance, it has been used to model the random behavior of asset returns.