# Difference between revisions of "Stochastic systems courses"

This page collects some information about stochastic systems courses offered at Caltech. This page was prepared in preparation for a faculty discussion on the current stochastic systems sequence (ACM/EE 116, ACM 216, ACM 217/EE 164).

### Overview of current course sequence

#### ACM 216: Markov Chain, Discret Stochastic Processes and Applications

##### Catalog listing

Stable laws, Markov chains, classification of states, ergodicity, von Neumann ergodic theorem, mixing rate, stationary/equilibrium distributions and convergence of Markov chains, Markov chain Monte Carlo and its applications to scientific computing, Metropolis Hastings algorithm, coupling from the past, martingale theory and discrete time martingales, rare events, law of large deviations, Chernoff bound

##### Topics (Winter 2008)
1. Markov Models.
2. Transition matrices and Markov Chains.
3. Kernels and Markov chains on arbitrary spaces.
4. Finite Markov Chains.
5. Markov Chains on countable state spaces.
6. Simulations with Markov Chains.
7. MCMC algortithms.
8. Simulated annealing.
9. The Propp-Wilson algorithm.
10. Sandwiching.
11. Rate of convergence

There are several advanced courses that build on ACM 116/216 and are offered on a semi-regular basis:

• EE 164 -
• ACM 217 - Stochastic Differential Equations and Applications
• ACM 217 -
• ACM 256 - Large Deviation Theory and Concentration Inequalities

### Additional stochastic systems courses at Caltech

The following table lists all of the courses that I was able to find that have been taught in the last four years. Enrollments (when given) are for 2005-2008, based on data from the registrar.

### Course listings

The course listings below are from the Caltech catalog, mainly to serve as a reference for the rest of the information on this page.

<span id=ACM116 /> ACM/EE 116. Introduction to Stochastic Processes and Modeling. 9 units (3-0-6); first term. Prerequisite: Ma 2 ab or instructor’s permission.Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Random variables, expectation and conditional expectation, joint distributions, covariance, moment generating function, central limit theorem, weak and strong laws of large numbers, discrete time stochastic processes, stationarity, power spectral densities and the Wiener-Khinchine theorem, Gaussian processes, Poisson processes, Brownian motion. The course develops applications in selected areas such as signal processing (Wiener filter), information theory, genetics, queuing and waiting line theory, and finance.

<span id=ACM118 /> ACM/ESE 118. Methods in Applied Statistics and Data Analysis. 9 units (3-0-6); first term. Prerequisite: Ma 2 or another introductory course in probability and statistics. Introduction to fundamental ideas and techniques of statistical modeling, with an emphasis on conceptual understanding and on the analysis of real data sets. Multiple regression: estimation, inference, model selection, model checking. Regularization of ill-posed and rank-deficient regression problems. Cross-validation. Principal component analysis. Discriminant analysis. Resampling methods and the bootstrap.

<span id=ACM216 /> ACM 216. Markov Chains, Discrete Stochastic Processes and Applications. 9 units (3-0-6); second term. Prerequisite: ACM/EE 116 or equivalent. Stable laws, Markov chains, classification of states, ergodicity, von Neumann ergodic theorem, mixing rate, stationary/equilibrium distributions and convergence of Markov chains, Markov chain Monte Carlo and its applications to scientific computing, Metropolis Hastings algorithm, coupling from the past, martingale theory and discrete time martingales, rare events, law of large deviations, Chernoff bounds.

<span id=ACM217 /> ACM 217. Advanced Topics in Stochastic Analysis. 9 units (3-0-6); third term. Prerequisite: ACM 216 or equivalent. The topic of this course changes from year to year and is expected to cover areas such as stochastic differential equations, stochastic control, statistical estimation and adaptive filtering, empirical processes and large deviation techniques, concentration inequalities and their applications. Examples of selected topics for stochastic differential equations include continuous time Brownian motion, Ito’s calculus, Girsanov theorem, stopping times, and applications of these ideas to mathematical finance and stochastic control. Not offered 2008–09.

<span id=ACM257 /> ACM 257. Special Topics in Financial Mathematics. 9 units (3-0-6); third term. Prerequisite: ACM 95/100 or instructor’s permission. A basic knowledge of probability and statistics as well as transform methods for solving PDEs is assumed. This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling. The mathematical concepts/tools developed will include introductions to random walks, Brownian motion, quadratic variation, and Ito-calculus. Connections to PDEs will be made by Feynman-Kac theorems. Concepts of risk-neutral pricing and martingale representation are introduced in the pricing of options. Topics covered will be selected from standard options, exotic options, American derivative securities, term-structure models, and jump processes. Instructor: Hill.

<span id=Ae115 /> Ae 115 ab. Spacecraft Navigation. 9 units (3-0-6); first, second terms. Prerequisite: CDS 110 a. This course will survey all aspects of modern spacecraft navigation, including astrodynamics, tracking systems for both low-Earth and deep-space applications (including the Global Positioning System and the Deep Space Network observables), and the statistical orbit determination problem (in both the batch and sequential Kalman filter implementations). The course will describe some of the scientific applications directly derived from precision orbital knowledge, such as planetary gravity field and topography modeling. Numerous examples drawn from actual missions as navigated at JPL will be discussed.

<span id=CDS110 /> CDS 110 ab. Introductory Control Theory. 12 units (3-0-9) first, 9 units (3-0-6) second terms. Prerequisites: Ma 1 and Ma 2 or equivalents; ACM 95/100 may be taken concurrently. An introduction to analysis and design of feedback control systems, including classical control theory in the time and frequency domain. Modeling of physical, biological, and information systems using linear and nonlinear differential equations. Stability and performance of interconnected systems, including use of block diagrams, Bode plots, the Nyquist criterion, and Lyapunov functions. Robustness and uncertainty management in feedback systems through stochastic and deterministic methods. Introductory random processes, Kalman filtering, and norms of signals and systems. The first term of this course is taught concurrently with CDS 101, but includes additional lectures, reading, and homework that is focused on analytical techniques for design and synthesis of control systems.

<span id=EE163 /> EE 163 ab. Communication Theory. 9 units (3-0-6); second, third terms. Prerequisite: EE 111; ACM/EE 116 or equivalent. Least mean square error linear filtering and prediction. Mathematical models of communication processes; signals and noise as random processes; sampling and quantization; modulation and spectral occupancy; intersymbol interference and synchronization considerations; signal-to-noise ratio and error probability; optimum demodulation and detection in digital baseband and carrier communication systems.

<span id=EE164 /> EE 164. Stochastic and Adaptive Signal Processing. 9 units (3-0-6); third term. Prerequisite: ACM/EE 116 or equivalent. Fundamentals of linear estimation theory are studied, with applications to stochastic and adaptive signal processing. Topics include deterministic and stochastic least-squares estimation, the innovations process, Wiener filtering and spectral factorization, state-space structure and Kalman filters, array and fast array algorithms, displacement structure and fast algorithms, robust estimation theory and LMS and RLS adaptive fields.

<span id=Ma112 /> Ma 112 ab. Statistics. 9 units (3-0-6); first, second terms. Prerequisite: Ma 2 a probability and statistics or equivalent. The first term covers general methods of testing hypotheses and constructing confidence sets, including regression analysis, analysis of variance, and nonparametric methods. The second term covers permutation methods and the bootstrap, point estimation, Bayes methods, and multistage sampling.

<span id=Ma114 /> Ma/ACM 144 ab. Probability. 9 units (3-0-6); second, third terms. Overview of measure theory. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Characteristic functions and the central limit theorem. Poisson process and Brownian motion.

<span id=Ma193 /> Ma 193 a. Random Matrix Theory. 9 units (3-0-6); first term. Prerequisite: Ma 108. Wigner matrices, Gaussian and circular ensembles of random matrices. Dyson's threefold way: orthogonal, unitary, and symplectic ensembles. Correlation functions; determinantal and Pfaffi an random point processes. Scaling limits. Fredholm determinant approach to gap probabilities.

<span id=SS214 /> SS/Ma 214. Mathematical Finance. 9 units (3-0-6); second term. A course on fundamentals of the mathematical modeling of stock prices and interest rates, the theory of option pricing, risk management, and optimal portfolio selection. Students will be introduced to the stochastic calculus of various continuous-time models, including diffusion models and models with jumps.

<span id=SS228 /> SS 228. Applied Data Analysis for the Social Sciences. 9 units (3-0-6); third term. The course covers issues of management and computation in the statistical analysis of large social science databases. Maximum likelihood and Bayesian estimation will be the focus. This includes a study of Markov Chain Monte Carlo (MCMC) methods. Substantive social science problems will be addressed by integrating programming, numerical optimization, and statistical methodology.