Stochastic processes in engineering systems This book is a revision of Stochastic Processes in Information and Dynamical Systems written by the first author (E.W.) and published in 1971. The book was originally written and revised to provide a graduate level text in stochastic processes for students whose primary interest is its applications. It treats both the traditional topic of sta­ tionary processes in linear time invariant systems as well as the more modern theory of stochastic systems in which dynamic structure plays a profound role. Our aim is to provide a high level yet readily acces­ sible treatment of those topics in the theory of continuous parameter stochastic processes that are important in the analysis of information and dynamical systems. The theory of stochastic processes can easily become abstract. In dealing with it from an applied point of view we have found it difficult to decide on the appropriate level of rigor. We intend to provide just enough mathematical machinery so that important results can be stated PREFACE vi with precision and clarity; so much ofthe theory of stochastic processes is inherently simple if the suitable framework is provided. The price of providing this framework seems worth paying even though the ul­ timate goal is in applications and not the mathematics per se. 1 Elements of Probability Theory 1. Events and probability 2. Measures on finite dimensional spaces 3. Measurable functions and random variables 4. Sequences of events and random variables 5. Expectation of random variables 6. Convergence concepts 7. Independence and conditional expectation 2 Stochastic Processes 1. Definition and preliminary considerations 2. Separability and measurability 3. Gaussian processes and Brownian motion 4. Continuity 5. Markov processes 6. Stationarity and ergodicity 3 Second Order Processes 1. Introduction 2. Second order continuity 3. Linear operations and second order calculus 4. Orthogonal expansions 5. Wide sense stationary processes 6. Spectral representation 7. Lowpass and bandpass processes 8. White noise and white noise integrals 9. Linear prediction and filtering 4 Stochastic Integrals and Stochastic Differential Equations 1. Introduction 2. Stochastic integrals 3. Processes defined by stochastic integrals 4. Stochastic differential equations 5. White noise and stochastic calculus 6. Generalizations of the stochastic integral 7. Diffusion equations 5 One Dimensional Diffusions 1. Introduction 2. The Markov semigroup 3. Strong Markov processes 4. Characteristic operators 5. Diffusion processes 6 Martingale Calculus 1. Martingales 2. Sample path integrals 3. Predictable processes 4. Isometric integrals 5. Semimartingale integrals 6. Quadratic variation and the change of variable formula 7. Semimartingale exponentials and applications 7 Detection and Filtering 1. Introduction 2. Likelihood ratio representation 3. Filter representation—change of measure derivation 4. Filter representation—innovations derivation 5. Recursive estimation 8 Random Fields 1. Introduction 2. Homogenous random fields 3. Spherical harmonics and isotropic random fields 4. Markovian random fields 5. Multiparameter martingales 6. Stochastic differential forms References Solutions to Exercises.