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 representationchange of measure derivation. - 4. Filter representationinnovations derivation. - 5. Recursiveestimation. - 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. Language: English