Foundations
- Combinatorics
- Common distributions:
- Generating Functions
- Probability Fundamentals:
- Conditional Probability
- Independence
- Bayes Theorem
- Law of Total Probability
- Convex and Concave Functions
Measure-Theoretic Probability
- Measure Theory Notes
- Measure Spaces
- Sample spaces, Events, Probability Spaces
- σ-algebras and Measurable spaces
- Probability measures and axioms (Kolmogorov’s axioms)
- Random Variables and Distributions
- Measurable functions, distribution functions
- Expectation and Moments
- Lebesgue Integrals
- Law of the unconscious statistician (LOTUS)
- Variance, covariance, higher-order moments
- Integral Results
- Random Variable Convergence
- Convergence Almost Surely, Mean, Probability, Law
- Borel-Cantelli Lemma
- Uniform Integrability
- Skorohod’s Representation Theorem
- Convergence Theorems
- Monotone Convergence Theorem
- Dominated Convergence Theorem
- Lp Spaces
- Limit Theorems
- Large Deviations
- Cramér Theorem
- Sanov Theorem
- Functional Limit Theorems
- Donsker Invariance Principle
- Prohorov Theorem (tightness)
- Conditional Expectation
- Hilbert Spaces
- Projection Theorem
Discrete-Time Stochastic Processes
- Markov Chains 1
- Transition matrices
- Chapman-Kolmogorov equations
- Classification of states (recurrent, transient, ergodic)
- Stationary distributions and convergence
- Martingale Theory
- Filtrations & Adapted Processes
- Martingale convergence theorem
- Optional stopping theorem
- Doob’s decomposition and inequalities
- Renewal Theory
- Renewal function, renewal reward processes
- Elementary and key renewal theorems
- Applications: queues, reliability theory
- Branching Processes
- Galton-Watson Branching Process
- Extinction Probabilities
- Random Walks
Continuous-Time Stochastic Processes
- Poisson Processes
- Inter-arrival times, memoryless property
- Compound Poisson processes
- Continuous-Time Markov Chains
- Infinitesimal generator,
- Kolmogorov
- Kolmogorov Forward Equation
- Kolmogorov Backward Equation
- Birth-death processes
- Brownian Motion
- Construction and properties (scaling, Markov, Gaussianity)
- Reflection principle, hitting times
- Functional CLT → Brownian motion as scaling limit
Stochastic Calculus
- Stochastic Calculus
- Itô Integration
- Itô isometry
- Itô’s Lemma
- Multidimensional version
- Change-of-variable formula
- P-Variation
- Stochastic Differential Equations (SDEs)
- Existence and uniqueness theorems
- Fokker–Planck Equation
- Kolmogorov PDEs
- Martingale Representation Theorem
- Common SDEs:
- Girsanov’s Theorem
- Change of measure
- Radon–Nikodym derivative
- Risk-neutral pricing in finance
- Optimal Stopping and Control
- Snell envelope
- Hamilton–Jacobi–Bellman equations
- Dynamic programming principle
- Mean Field Games and McKean-Vlasov SDEs
- Nash equilibria in continuum agent models
- Forward-backward SDEs
Advanced Topics
- Interacting Particle Systems and Mean Field Models
- Propagation of chaos
- McKean–Vlasov equations
- Queuing Theory
- M/M/1, M/G/1, G/G/1 queues
- Little’s Law, heavy traffic approximations
- Large Deviations and Rare Event Simulation
- Importance sampling, importance splitting
- Freidlin–Wentzell theory for SDEs
- Connections with PDEs
- Probabilistic representations (Feynman–Kac, Dynkin’s formula)
- Stochastic representations for solutions to elliptic and parabolic PDEs
- Measure-Valued and Distribution-Valued Processes
- Superprocesses, stochastic PDEs
- Applications in population dynamics and filtering
- Stochastic Partial Differential Equations (SPDEs)
- Regularity structures (Hairer)
- Paracontrolled calculus
- Stochastic Topology and Random Geometry
- Stochastic homology, random graphs, and geometric complexes
- Interplay with Functional Analysis
- Dirichlet forms, Gaussian Hilbert spaces
- Infinite-dimensional stochastic analysis (Malliavin calculus)