← All courses

Stochastic Processes and Martingales

Year 3 · 5 sections · 52 concepts · 90 practice questions

Start free (beta)Try chapter 1 free, no sign-up →

What's covered

free1 · Conditional expectation9 concepts

Overview 1 · The plan of this section
Definition 1.1 · Information as a σ-algebra
Definition 1.2 · Conditional expectation
Proposition 1.3 · Uniqueness of conditional expectation
Theorem 1.4 · Rules for conditional expectation
Proposition 1.5 · Conditional Jensen inequality
Theorem 1.6 · Conditional expectation as orthogonal projection
Example 1.7 · Computing conditional expectations
Check 1.8 · Check: computing E[X | G]

2 · Martingales and optional stopping12 concepts

Overview 2 · The plan of this section
Definition 2.1 · Filtrations and adapted processes
Definition 2.2 · Martingales, submartingales and supermartingales
Example 2.3 · Standard martingales
Proposition 2.4 · Martingale transforms (you can't beat the system)
Definition 2.5 · Stopping times
Proposition 2.6 · Stopped martingales are martingales
Theorem 2.7 · The optional stopping theorem
Example 2.8 · When optional stopping fails
Example 2.9 · Gambler's ruin by martingales
Corollary 2.10 · Wald's identity
Check 2.11 · Check: using optional stopping

3 · Martingale convergence11 concepts

Overview 3 · The plan of this section
Theorem 3.1 · Doob's maximal inequality
Corollary 3.2 · Doob's L² inequality
Lemma 3.3 · The upcrossing lemma
Theorem 3.4 · The martingale convergence theorem
Corollary 3.5 · Non-negative supermartingales converge
Theorem 3.6 · L²-bounded martingales
Example 3.7 · Pólya's urn
Example 3.8 · The branching process martingale
Remark 3.9 · Uniform integrability and L¹ convergence
Check 3.10 · Check: proving a process converges

4 · Continuous-time Markov chains10 concepts

Overview 4 · The plan of this section
Definition 4.1 · Generators and jump chains
Theorem 4.2 · The transition semigroup and Kolmogorov's equations
Example 4.3 · The two-state chain
Theorem 4.4 · Stationary distributions
Proposition 4.5 · Detailed balance
Example 4.6 · Birth–death chains
Example 4.7 · The M/M/1 queue
Proposition 4.8 · Expected hitting times
Check 4.9 · Check: analysing a continuous-time chain

5 · Brownian motion10 concepts

Overview 5 · The plan of this section
Definition 5.1 · Brownian motion
Proposition 5.2 · Brownian motion as a Gaussian process
Proposition 5.3 · Invariances of Brownian motion
Proposition 5.4 · Brownian martingales
Example 5.5 · Exit from an interval
Theorem 5.6 · The reflection principle
Proposition 5.7 · Quadratic variation
Remark 5.8 · Brownian motion as a scaling limit
Check 5.9 · Check: Brownian computations

Try it: a free card from this module

This is exactly what studying looks like inside UniMaths.

Proposition 1.3 · Uniqueness of conditional expectation

The defining property pins $\mathbb E[X\mid\mathcal G]$ down almost surely.

Proof

Learn all 52 concepts like this

Cards, step-by-step proofs, and 90 questions scheduled so you review right before you forget.

Start free (beta)Try chapter 1 first