Stochastic Processes and Simulation

What this page covers

The resources here fall into two broad groups: the theory of random processes, and simulation. On the theory side you will find mind maps on Markov chains, the Chapman and Kolmogorov equations, and the Birth and Death Process, plus flashcard decks on Poisson processes and Brownian motion. On the simulation side there are materials on Monte Carlo methods and on random number generators.

Where to start

If Markov chains are new, begin with the Markov chains mind map. It sets out the Markov property, that the next step depends only on the immediately previous event, and the transition probabilities. Then move to the mind map on the Equations of Chapman and Kolmogorov, which relates states and probabilities over several periods.

1. Read the Markov chains mind map.
2. Follow with the Chapman and Kolmogorov equations.
3. Move on to the Birth and Death Process, which covers rates of arrival and exit.
4. Test yourself with the flashcards on Poisson processes.

Simulation and random numbers

The Monte Carlo simulation mind map explains where the technique comes from and what it is used for. The Automatic Random Variate Generation mind map covers fast, black box generators that work for large classes of distributions. Two quizzes then let you check your understanding: QUIZ 7 asks about pseudo-random number generators, seeds and the Monte Carlo method, and SM-Random Number Generator tests the idea of a pseudorandom generator through fill-in-the-blank questions.

Using quizzes, maps and flashcards together

Read a map first to see how the ideas connect. Take a quiz next to find what you missed, then go back to that branch of the map. Flashcards suit definitions, such as the ones in the Probability: Definitions deck, and the filtration and martingale terms in the Chapter 4 Brownian motion deck. Teachers can use any of these as ready-made material for a class or a revision handout.

Where this unit sits

This unit belongs to the Probability strand of Statistics and probability. Pick one resource that matches what you are studying this week and work through it.