Stochastic processes
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Probability foundations
- Review of measure theory: σ-algebras, measures, and integration
- Random variables, distributions, and expectation
- Conditional expectation and filtrations
- Convergence modes: a.s., in probability, Lp, and distribution
- Characteristic functions and continuity theorems (introduction)
Discrete-time processes
- Markov chains: classification of states and stationary distributions
- Martingales: definitions, stopping times, and optional stopping
- Doob's martingale convergence theorem (statement)
- Martingale inequalities: Doob and Burkholder (introduction)
- Random walks and renewal theory (introduction)
Continuous-time processes
- Poisson processes and compound Poisson processes
- Brownian motion: construction and path properties
- Itô integral and Itô's lemma (introduction)
- Stochastic differential equations and existence/uniqueness (overview)
- Markov property and generators (introduction)
STEM / applied
Applications in finance and engineering
- Geometric Brownian motion and Black–Scholes (mathematical setup)
- Queueing theory and Markovian service models
- Filtering and Kalman filter (introduction)
- Monte Carlo simulation of SDEs
- Risk measures and value-at-risk (overview)
Computational and statistical methods
- Estimation for stochastic models (MLE, method of moments)
- Simulation of Markov chains and point processes
- Time series as stochastic processes (ARMA connection)
- Numerical methods for SDEs: Euler–Maruyama and Milstein
- Case studies in biology, physics, and operations research
Notes
Topics reflect common graduate stochastic processes syllabi at US universities. Some programs split discrete and continuous-time material across two courses.