Stochastic Processes

What you will learn

Stochastic processes, also called random processes, are random phenomena that depend on time. In this module, you will learn the basics of stochastic processes with some applications. 

Prerequisites

I assume you are familiar with basic calculus, some linear algebra, and elementary probability and statistics (mostly what you learn in the first one or two years in university). I do not assume the knowledge of measure-theoretic probability theory (if you don't know what that is, never mind). As such, the exposition here is relatively informal and lacks mathematical rigor. Nevertheless, I believe this is sufficiently useful for many applications in science and engineering. It also serves as the first step to more elaborate and rigorous theories of stochastic processes.

Lecture Notes

This page will contain the lecture notes on Stochastic Processes.

  1. Review of probability theory
  2. Gambler's ruin problem
  3. Random walks
    1. Simulating random walks
  4. Markov chains
  5. Poisson process
    1. Introduction
    2. Differential-difference equations
    3. Derivation
    4. Solving differential-difference equations with an iterative method
    5. Generating function
    6. Arrival times
    7. Simulating Poisson processes
  6. Birth and death processes
    1. Birth process
    2. Death process
    3. Combining birth and death processes
    4. Simulating birth processes
    5. Simulating death processes
    6. Simulating birth-death processes
  7. Queues
  8. Reliability and renewal processes
  9. Branching processes
    1. Mean and variance
    2. Probability of extinction
    3. Martingales and stopping time
  10. Epidemic models
    1. Simple epidemic models
  11. Brownian motion (Wiener processes)
See also: the YouTube playlist Stochastic Processes

References

Most of the above contents are based on Jones and Smith (2018).

  1. P. W. Jones and P. Smith (2018), Stochastic Processes, (3rd Ed.), CRC Press.

  2. G. R. Grimmett and D. R. Stirzaker (2001), Probability and Random Processes (3rd Ed.), Oxford University Press.

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