Introduction to Random Processes

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). 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. Front Matter....Pages I-VIII Random Processes with Discrete State Space....Pages 1-9 Homogeneous Markov Processes with a Countable Number of States....Pages 10-17 Homogeneous Markov Processes with a Countable Number of States....Pages 18-24 Branching Processes....Pages 25-32 Brownian Motion....Pages 33-43 Random Processes in Multi-Server Systems....Pages 44-51 Random Processes as Functions in Hilbert Space....Pages 52-56 Stochastic Measures and Integrals....Pages 57-60 The Stochastic Ito Integral and Stochastic Differentials....Pages 61-67 Stochastic Differential Equations....Pages 68-72 Diffusion Processes....Pages 73-76 Linear Stochastic Differential Equations and Linear Random Processes....Pages 77-83 Stationary Processes....Pages 84-91 Some Problems of Optimal Estimation....Pages 92-99 A Filtration Problem....Pages 100-117 Back Matter....Pages 108-120