Introduction

Stochastic processes is the mathematical theory of random phenomena. Our working definition of a random phenomenon or process, is one that we cannot predict accurately. Physical processes are often not predictable for a variety of reasons. Typically, we quantify a physical process somehow, e.g. by taking measurement at specific times. If each measurement is unpredicatable (random), then our sequence of measurements is a stochastic process!

You have seen random variables in previous probability or statistics courses. A random variable \(X\) takes on real number values, but we cannot predict what precise value it will take perfectly… it is random. One can think of performing a random experiment such as rolling a die and letting \(X\) be the number of dots on the upper face or observing some physical process like drilling an oil well with \(X\) being the amount of oil produced on the first day or growing a plant and letting \(X\) be the height of the plant after one month of growth. In all of the examples \(X\) just takes on a single numerical value. A stochastic process tracks these processes over time. Let \(X_n\) be the outcome of the \(n^{th}\) die roll, the height of the plant after month \(n\), or the amount of oil produced during day \(n\). In this way a stochastic process can initially be thought of as a sequence of random variables.

Here are some examples of how a stochastic process might model a physical process.

  1. Modeling the daily closing price of a stock for one year.

  2. Modeling the number of insurance claims in each month over a year.

  3. The number of new infections each day for a particular disease.

  4. Tracking radioactive decays over time.

  5. The location of an animal as it moves through its habitat, e.g. the distance from a bird to its nest as a funciton of time.

Each of these physical phenomena are highly unpredictable, and so we generally treat them as “random.”