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Today, the theory of random processes represents a large field of mathematics with many different branches, and the task of choosing topics for a brief introduction to this theory is far from being simple. This introduction to the theory of random processes uses mathematical models that are simple, but have some importance for applications. We consider different processes, whose development in time depends on some random factors. The fundamental problem can be briefly circumscribed in the following way: given some relatively simple characteristics of a process, compute the probability of another event which may be very complicated; or estimate a random variable which is related to the behaviour of the process. The models that we consider are chosen in such a way that it is possible to discuss the different methods of the theory of random processes by referring to these models. The book starts with a treatment of homogeneous Markov processes with a countable number of states. The main topic is the ergodic theorem, the method of Kolmogorov's differential equations (Secs. 1-4) and the Brownian motion process, the connecting link being the transition from Kolmogorov's differential-difference equations for random walk to a limit diffusion equation (Sec. 5).
Table des matières
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.
