Standard syllabus
Stochastic processes · Graduate · Math
Learning objectives from the Stochastic processes syllabus, grouped by unit. Click an objective for study materials.
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
Learning objectives
Click an objective for study materials.
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
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
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)
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
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
Definitions and structure
- 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
Proofs and reasoning
- Convergence modes — Convergence modes
- Distribution — Distribution
- Characteristic functions — Characteristic functions
- Continuity theorems (introduction) — Continuity theorems (introduction)
- Markov chains — Markov chains
Abstraction and generalization
- 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
Modeling and computation
- Apply review of measure theory in engineering contexts — Apply review of measure theory in engineering contexts
- Apply integration in engineering contexts — Apply integration in engineering contexts
- Apply random variables and distributions in engineering contexts — Apply random variables and distributions in engineering contexts
- Apply expectation in engineering contexts — Apply expectation in engineering contexts
- Apply conditional expectation and filtrations in engineering contexts — Apply conditional expectation and filtrations in engineering contexts
Data and technology
- Use software to explore stochastic processes problems numerically — Use software to explore stochastic processes problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Convergence modes — Convergence modes
- Distribution — Distribution
- Characteristic functions — Characteristic functions
- Continuity theorems (introduction) — Continuity theorems (introduction)
- Markov chains — Markov chains
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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$1,162 · Stochastic processes · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.