Standard syllabus
Stochastic processes · Graduate · Math
Topics
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)
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$1,162 · Stochastic processes · 18 tutoring hrs
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