Introduction to Random Processes

ISBN-13: 9783642727191 / Angielski / Miękka / 2011 / 117 str.

Yurii A. Rozanov; Birgit R. Thinger

Opis

Today, the theory of random processes represents a large field of mathematics with many different branches. This "Introduction to the Theory of Random Processes" applies mathematical models that are simple, but that have some importance for applications. The book starts with a treatment of homogeneous Markov processes with a countable number of states. The main topics are the ergodic theorem, the method of Kolmogorov's differential equations and Brownian motion, and the connecting link being the transition from Kolmogorov's differential-difference equations for random walk to a limit diffusion equation. The chapters that follow outline the foundations of stochastic analysis. They deal with random processes as curves in the space of random variables with the norm of quadratic mean. Random processes are then described by linear stochastic differential equations and their convergence behaviour is explored. The fundamentals of spectral analysis of stationary processes are considered and, finally, some special problems of estimation and filtration are discussed. In chapter 6 an attempt is made to apply direct probabilistic methods for sums of i.i.d. variables to a multi-server-system. As a complement, chapters 9 to 11 deal with nonlinear stochastic differential equations for diffusion processes.

Spis treści

Section 1. Random Processes with Discrete State Space. Examples.- Section 2. Homogeneous Markov Processes with a Countable Number of States. Kolmogorov’s Differential Equations.- Section 3. Homogeneous Markov Processes with a Countable Number of States. Convergence to a Stationary Distribution.- Section 4. Branching Processes. Method of Generating Functions.- Section 5. Brownian Motion. The Diffusion Equation and Some Properties of the Trajectories.- Section 6. Random Processes in Multi-Server Systems.- Section 7. Random Processes as Functions in Hilbert Space.- Section 8. Stochastic Measures and Integrals.- Section 9. The Stochastic Ito Integral and Stochastic Differentials.- Section 10. Stochastic Differential Equations.- Section 11. Diffusion Processes. Kolomogorov’s Differential Equations.- Section 12. Linear Stochastic Differential Equations and Linear Random Processes.- Section 13. Stationary Processes. Spectral Analysis and Linear Transformations.- Section 14. Some Problems of Optimal Estimation.- Section 15. A Filtration Problem. Kalman-Bucy Filter.- Appendix. Basic Concepts of Probability Theory.