Course Outline
This course does not assume a measure-theoretic probability background. The connections between stochastic analysis and linear algebra are emphasized, and the Monte Carlo method as a numerical tool is introduced and repeatedly discussed within the various applications that are covered.
The Monte Carlo Method
- Computer experimentation
- Convergence rate
- Confidence intervals
- Advantages/disadvantages
- Bootstrap
Linear and Gaussian Models
- Regression
- Least squares viewpoint
- Statistical viewpoint and advantages of statistical formulation
- Maximum likelihood
- Bayesian formulation
- Prediction:
- Hilbert space of square-integrable rvs
- Prediction theory
- Gaussian prediction theory and related linear algebra
- Autoregressive modeling/state space models and prediction
- Filtering:
- Kalman filtering
- Gaussian processes and fields:
- Gaussian processes (covariance function; continuity/differentiability)
- Gaussian random fields
Discrete Time Markov Chains
- Transient behavior
- First-step analysis
- Equilibrium behavior
- Stochastic control/dynamic programming (HJB equation),
- Markov chain Monte Carlo
Continuous-Time Markov Chains
- Transient behavior
- Equilibrium behavior
- Reversibility
- Product-form networks
- Thermodynamic limits
Diffusions
- Brownian motion
- Stochastic differential equations (SDEs)
- Connection between SDEs and PDEs
- Numerical solution of SDE's (Euler method, Milstein method)