Stochastic Processes & Martingales

Poisson Process Terminology (X_i, Y_i, J_i, S_i)

Poisson Process

2 Types of Increments

How to answer Poisson Process question (e.g. P(X₇=3), \lambda=¹/₂)

Sum of two Poisson Processes

Colouring Theorem

Poisson Point Process

Compound Poisson Process

Mean and Variance of a Compound Poisson Process

Central Limit Theorem (used for CPPs with large n e.g. P(X₅₀>240))

Memoryless Property

Test for Explosion of a Birth Process

Mean and Variance of Birth Process

Irreducibility of Markov Chain

Transient, Null/Positive Recurrent Definitions

How to work out expected first arrival time at node i

Long term probability, P_i(X_t = j)

Detailed Balance

Find Invariant Distribution

Long run probability of something happening (in Markov Chain)

Stationary Distribution for Closed Migration Process

Power Set

Probability Space

\sigma-algebra on a set \Omega

F-measurable

Filtration

\sigma(X)

Natural Filtration of (X_n)

Properties of the Expectation

Properties of the Conditional Expectation

How to define a Martingale

Stopping Time T

Optional (and Elementary) Stopping Theorem

Martingale Convergence Theorem