**ELE 529 - Theoretical Foundations of Random
Processes **

Spring 1985-86

Professor Sergio Verdú

**Prerequisites:** Engineering background on random processes equivalent
to EE525

**Textbook: **E. Wong and B. Hajek, Stochastic Processes in Engineering
Systems, Springer-Verlag 1985.

**I. Probability Theory**

Probability space - Measurable functions - Expectation - Sequences of random variab les and convergence -Independence

**II. Conditioning. Discrete-time processes: Markov property and martingales**

Conditional expectation - Stopping times - Markov property - Martingales - Convergence and inequalities

**III. Continuous-time processes --fundamentals**

Kolmogorov Existence theorem - Separability and measurability - Continuity - Statio narity and Ergodicity - Brownian motion. Poisson processes

**IV. Stochasic integration and stochastic differential equations**

Stochastic integral. It o calculus - Uniqueness and existence of solutions of stochastic differential equations - Diffusion processes.

**V. Applications in detection, estimation and stochastic control**

Girsanov's transformation - Detection of Stochastic Signals - Existence and characterization of optimal stochastic control laws

**REFERENCES**

R. Ash, Real Analysis and Probability , New York: Academic, 1972

R. Ash and M.F. Gardner, Topics in Stochastics Processes, New York: Academic, 1975

P. Billingsley, Probability and Measure , Wiley, 1979 K.L. Chung, A Course in Probability Theory , Second Ed. New York: Academic, 1974

J.L. Doob, Stochastic Processes , Wiley, 1953

A. Friedman, Stochastic Differential Equations and Applications, vol. 1. New York: Academic, 1975

V. Krishnan, Nonlinear filtering and smoothing: An introduction to martingales , stochastic integrals and estimation, Wiley, 1984

D.L. Snyder, Random Point Processes , Wiley, 1975

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