Point Processes and Queues: Martingale Dynamics.

Abstract

I Martingales.- 1. Histories and Stopping Times.- 2. Martingales.- 3. Predictability.- 4. Square-Integrable Martingales.- References.- Solutions to Exercises, Chapter I.- II Point Processes, Queues, and Intensities.- 1. Counting Processes and Queues.- 2. Watanabe's Characterization.- 3. Stochastic Intensity, General Case.- 4. Predictable Intensities.- 5. Representation of Queues.- 6. Random Changes of Time.- 7. Cryptographic Point Processes.- References.- Solutions to Exercises, Chapter II.- III Integral Representation of Point-Process Martingales.- 1. The Structure of Internal Histories.- 2. Regenerative Form of the Intensity.- 3. The Representation Theorem.- 4. Hilbert-Space Theory of Poissonian Martingales.- 5. Useful Extensions.- References.- Solutions to Exercises, Chapter III.- IV Filtering.- 1. The Theory of Innovations.- 2. State Estimates for Queues and Markov Chains.- 3. Continuous States and Nontrivial Prehistory.- References.- Solutions to Exercises, Chapter IV.- V Flows in Markovian Networks of Queues.- 1. Single Station : The Historical Results and the Filtering Method.- 2. Jackson's Networks.- 3. Burke's Output Theorem for Networks.- 4. Cascades and Loops in Jackson's Networks.- 5. Independence and Poissonian Flows in Markov Chains.- References.- Solutions to Exercises, Chapter V.- VI Likelihood Ratios.- 1. Radon-Nikodym Derivatives and Tests of Hypotheses.- 2. Changes of Intensities la Girsanov.- 3. Filtering by the Method of the Probability of Reference.- 4. Applications.- 5. The Capacity of a Point-Process Channel.- 6. Detection Formula187 References189 Solutions to Exercises, Chapter VI.- VII Optimal Control.- 1. Modeling Intensity Controls.- 2. Dynamic Programming for Intensity Controls : Complete-Observation Case.- 3. Input Regulation. A Case Study in Impulsive Control.- 4. Attraction Controls.- 5. Existence via Likelihood Ratio.- References.- Solutions to Exercises, Chapter VII.- VIII Marked Point Processes.- 1. Counting Measure and Intensity Kernels.- 2. Martingale Representation and Filtering.- 3. Radon-Nikodym Derivatives.- 4. Towards a General Theory of Intensity.- References.- Solutions to Exercises, Chapter VIII.- A1 Background in Probability and Stochastic Processes.- 1. Introduction.- 2. Monotone Class Theorem.- 3. Random Variables.- 4. Expectations.- 5. Conditioning and Independence.- 6. Convergence.- 7. Stochastic Processes.- 8. Markov Processes.- References.- A2 Stopping Times and Point-Process Histories.- 1. Stopping Times.- 2. Changes of Time and Meyer-Dellacherie's Integration Formula.- 3. Point-Process Histories.- References.- A3 Wiener-Driven Dynamical Systems.- 1. Ito's Stochastic Integral.- 2. Square-Integrable Brownian Martingales.- 3. Girsanov's Theorem.- References.- A4 Stieltjes-Lebesgue Calculus.- 1. The Stieltjes-Lebesgue Integral.- 2. The Product and Exponential Formulas.- References.- General Bibliography.