Introduction to Stochastic Processes

Abstract

In financial modeling, practitioners draw upon tools in the field of mathematics. In asset return modeling, option pricing modeling, term structure modeling, and credit risk modeling, the primary mathematical tool used by practitioners is stochastic processes which concerns sequences of events that are governed by the laws of probability. When the sequence of events for a stochastic process involves data points measured at successive time intervals, the sequence is referred to as a time series. Practitioners seek to understand the behavior of a financial time series so that they can forecast or predict future returns based on past returns. Stochastic processes involving time series can be classified as discrete-time stochastic processes and continuous-time stochastic processes. Keywords: stochastic process; discrete-time stochastic process; continuous-time stochastic process; autoregressive moving average; white noise process; autoregressive process; autocorrelation function; partial autocorrelation function (PACF) moving average process; general linear time series model; ARMAX processes; generalized autoregressive conditional heteroskedasticity (GARCH) process; autoregressive conditional heteroskedasticity (ARCH) process; Poisson process; Brownian motion; stochastic differential equations; jump term; Levy process; α-Stable Levy motion