Stochastic processes
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Stochastic Processes — scope drawn from open stochastic processes notes and typical US graduate stochastic processes syllabi.
Probability foundations
- Review of measure theory — Review of measure theory
- Integration — Integration
- Random variables and distributions — Random variables and distributions
- Expectation — Expectation
- Conditional expectation and filtrations — Conditional expectation and filtrations
- Convergence modes — Convergence modes
- Distribution — Distribution
- Characteristic functions — Characteristic functions
- Continuity theorems (introduction) — Continuity theorems (introduction)
Discrete-time processes
- Markov chains — Markov chains
- Stationary distributions — Stationary distributions
- Martingales — Martingales
- Optional stopping — Optional stopping
- Doob's martingale convergence theorem (statement) — Doob's martingale convergence theorem (statement)
- Martingale inequalities — Martingale inequalities
- Burkholder (introduction) — Burkholder (introduction)
- Random walks — Random walks
- Renewal theory (introduction) — Renewal theory (introduction)
Continuous-time processes
- Poisson processes — Poisson processes
- Compound Poisson processes — Compound Poisson processes
- Brownian motion — Brownian motion
- Path properties — Path properties
- Itô integral — Itô integral
- Itô's lemma (introduction) — Itô's lemma (introduction)
- Stochastic differential equations — Stochastic differential equations
- Existence/uniqueness (overview) — Existence/uniqueness (overview)
- Markov property — Markov property
- Generators (introduction) — Generators (introduction)
Applications in finance and engineering
- Geometric Brownian motion — Geometric Brownian motion
- Black–Scholes (mathematical setup) — Black–Scholes (mathematical setup)
- Queueing theory — Queueing theory
- Markovian service models — Markovian service models
- Filtering — Filtering
- Kalman filter (introduction) — Kalman filter (introduction)
- Monte Carlo simulation of SDEs — Monte Carlo simulation of SDEs
- Risk measures — Risk measures
- Value-at-risk (overview) — Value-at-risk (overview)
Computational and statistical methods
- Estimation for stochastic models (MLE, method of moments) — Estimation for stochastic models (MLE, method of moments)
- Simulation of Markov chains — Simulation of Markov chains
- Point processes — Point processes
- Time series as stochastic processes (ARMA connection) — Time series as stochastic processes (ARMA connection)
- Numerical methods for SDEs: Euler–Maruyama and Milstein — Numerical methods for SDEs: Euler–Maruyama and Milstein
- Case studies in biology and physics — Case studies in biology and physics
- Operations research — Operations research
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Probability foundations
Review of measure theory
Coming soon - Discrete-time processes
Markov chains
Coming soon - Continuous-time processes
Poisson processes
Coming soon - Applications in finance and engineering
Geometric Brownian motion
Coming soon - Computational and statistical methods
Estimation for stochastic models (MLE, method of moments)
Coming soon
Notes
Graduate Stochastic Processes — scope drawn from open stochastic processes notes and typical US graduate stochastic processes syllabi. Topic outline: `content/topics/graduate/stochastic_processes.json`.