Stochastic Differential Equations for Science and Engineering

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Cover Half Title Title Page Copyright Page Contents Preface CHAPTER 1: Introduction SECTION I: Fundamentals CHAPTER 2: Diffusive Transport and Random Walks 2.1. DIFFUSIVE TRANSPORT 2.1.1. The Conservation Equation 2.1.2. Fick’s Laws 2.1.3. Diffusive Spread of a Point Source 2.1.4. Diffusive Attenuation of Waves 2.2. ADVECTIVE AND DIFFUSIVE TRANSPORT 2.3. DIFFUSION IN MORE THAN ONE DIMENSION 2.4. RELATIVE IMPORTANCE OF ADVECTION AND DIFFUSION 2.5. THE MOTION OF A SINGLE MOLECULE 2.6. MONTE CARLO SIMULATION OF PARTICLE MOTION 2.7. CONCLUSION 2.8. EXERCISES CHAPTER 3: Stochastic Experiments and Probability Spaces 3.1. STOCHASTIC EXPERIMENTS 3.2. RANDOM VARIABLES 3.3. EXPECTATION IS INTEGRATION 3.4. INFORMATION IS A σ-ALGEBRA 3.5. CONDITIONAL EXPECTATIONS 3.5.1. Properties of the Conditional Expectation 3.5.2. Conditional Distributions and Variances 3.6. INDEPENDENCE AND CONDITIONAL INDEPENDENCE 3.7. LINEAR SPACES OF RANDOM VARIABLES 3.8. CONCLUSION 3.9. NOTES AND REFERENCES 3.9.1. The Risk-Free Measure 3.9.2. Convergence for Sequences of Events 3.9.3. Convergence for Random Variables 3.10. EXERCISES CHAPTER 4: Brownian Motion 4.1. STOCHASTIC PROCESSES AND RANDOM FUNCTIONS 4.2. DEFINITION OF BROWNIAN MOTION 4.3. PROPERTIES OF BROWNIAN MOTION 4.4. FILTRATIONS AND ACCUMULATION OF INFORMATION 4.5. THE MARTINGALE PROPERTY 4.6. CONCLUSION 4.7. NOTES AND REFERENCES 4.8. EXERCISES CHAPTER 5: Linear Dynamic Systems 5.1. LINEAR SYSTEMS WITH DETERMINISTIC INPUTS 5.2. LINEAR SYSTEMS IN THE FREQUENCY DOMAIN 5.3. A LINEAR SYSTEM DRIVEN BY NOISE 5.4. STATIONARY PROCESSES IN TIME DOMAIN 5.5. STATIONARY PROCESSES IN FREQUENCY DOMAIN 5.6. THE RESPONSE TO NOISE 5.7. THE WHITE NOISE LIMIT 5.8. INTEGRATED WHITE NOISE IS BROWNIAN MOTION 5.9. LINEAR SYSTEMS DRIVEN BY WHITE NOISE 5.10. THE ORNSTEIN-UHLENBECK PROCESS 5.11. THE NOISY HARMONIC OSCILLATOR 5.12. CONCLUSION 5.13. EXERCISES SECTION II: Stochastic Calculus CHAPTER 6: Stochastic Integrals 6.1. ORDINARY DIFFERENTIAL EQUATIONS DRIVEN BY NOISE 6.2. SOME EXEMPLARY EQUATIONS 6.2.1. Brownian Motion with Drift 6.2.2. The Double Well 6.2.3. Geometric Brownian Motion 6.2.4. The Stochastic van der Pol Oscillator 6.3. THE ITÔ INTEGRAL AND ITS PROPERTIES 6.4. A CAUTIONARY EXAMPLE: ∫T0 BS dBS 6.5. ITÔ PROCESSES AND SOLUTIONS TO SDES 6.6. RELAXING THE L2 CONSTRAINT 6.7. INTEGRATION WITH RESPECT TO ITÔ PROCESSES 6.8. THE STRATONOVICH INTEGRAL 6.9. CALCULUS OF CROSS-VARIATIONS 6.10. CONCLUSION 6.11. NOTES AND REFERENCES 6.11.1. The Proof of Itô Integrability 6.11.2. Weak Solutions to SDE’s 6.12. EXERCISES CHAPTER 7: The Stochastic Chain Rule 7.1. THE CHAIN RULE OF DETERMINISTIC CALCULUS 7.2. TRANSFORMATIONS OF RANDOM VARIABLES 7.3. ITÔ’S LEMMA: THE STOCHASTIC CHAIN RULE 7.4. SOME SDE’S WITH ANALYTICAL SOLUTIONS 7.4.1. The Ornstein-Uhlenbeck Process 7.5. DYNAMICS OF DERIVED QUANTITIES 7.5.1. The Energy in a Position-Velocity System 7.5.2. The Cox-Ingersoll-Ross Process and the Bessel Processes 7.6. COORDINATE TRANSFORMATIONS 7.6.1. Brownian Motion on the Circle 7.6.2. The Lamperti Transform 7.6.3. The Scale Function 7.7. TIME CHANGE 7.8. STRATONOVICH CALCULUS 7.9. CONCLUSION 7.10. NOTES AND REFERENCES 7.11. EXERCISES CHAPTER 8: Existence, Uniqueness, and Numerics 8.1. THE INITIAL VALUE PROBLEM 8.2. UNIQUENESS OF SOLUTIONS 8.2.1. Non-Uniqueness: The Falling Ball 8.2.2. Local Lipschitz Continuity Implies Uniqueness 8.3. EXISTENCE OF SOLUTIONS 8.3.1. Linear Bounds Rule Out Explosions 8.4. NUMERICAL SIMULATION OF SAMPLE PATHS 8.4.1. The Strong Order of the Euler-Maruyama Method for Geometric Brownian Motion 8.4.2. Errors in the Euler-Maruyama Scheme 8.4.3. The Mil’shtein Scheme 8.4.4. The Stochastic Heun Method 8.4.5. The Weak Order 8.4.6. A Bias/Variance Trade-off in Monte Carlo Methods 8.4.7. Stability and Implicit Schemes 8.5. CONCLUSION 8.6. NOTES AND REFERENCES 8.6.1. Commutative Noise 8.7. EXERCISES CHAPTER 9: The Kolmogorov Equations 9.1. BROWNIAN MOTION IS A MARKOV PROCESS 9.2. DIFFUSIONS ARE MARKOV PROCESSES 9.3. TRANSITION PROBABILITIES AND DENSITIES 9.3.1. The Narrow-Sense Linear System 9.4. THE BACKWARD KOLMOGOROV EQUATION 9.5. THE FORWARD KOLMOGOROV EQUATION 9.6. DIFFERENT FORMS OF THE KOLMOGOROV EQUATIONS 9.7. DRIFT, NOISE INTENSITY, ADVECTION, AND DIFFUSION 9.8. STATIONARY DISTRIBUTIONS 9.9. DETAILED BALANCE AND REVERSIBILITY 9.10. CONCLUSION 9.11. NOTES AND REFERENCES 9.11.1. Do the Transition Probabilities Admit Densities? 9.11.2. Eigenfunctions, Mixing, and Ergodicity 9.11.3. Reflecting Boundaries 9.11.4. Girsanov’s Theorem 9.11.5. Numerical Computation of Transition Probabilities 9.11.5.1. Solution of the Discretized Equations 9.12. EXERCISES SECTION III: Applications CHAPTER 10: State Estimation 10.1. RECURSIVE FILTERING 10.2. OBSERVATION MODELS AND THE STATE LIKELIHOOD 10.3. THE RECURSIONS: TIME UPDATE AND DATA UPDATE 10.4. THE SMOOTHING FILTER 10.5. SAMPLING TYPICAL TRACKS 10.6. LIKELIHOOD INFERENCE 10.7. THE KALMAN FILTER 10.7.1. Fast Sampling and Continuous-Time Filtering 10.7.2. The Stationary Filter 10.7.3. Sampling Typical Tracks 10.8. ESTIMATING STATES AND PARAMETERS AS A MIXED-EFFECTS MODEL 10.9. CONCLUSION 10.10. NOTES AND REFERENCES 10.11. EXERCISES CHAPTER 11: Expectations to the Future 11.1. DYNKIN’S FORMULA 11.2. EXPECTED EXIT TIMES FROM BOUNDED DOMAINS 11.2.1. Exit Time of Brownian Motion with Drift on the Line 11.2.2. Exit Time from a Sphere, and the Diffusive Time Scale 11.2.3. Exit Times in the Ornstein-Uhlenbeck Process 11.2.4. Regular Diffusions Exit Bounded Domains in Finite Time 11.3. ABSORBING BOUNDARIES 11.4. THE EXPECTED POINT OF EXIT 11.4.1. Does a Scalar Diffusion Exit Right or Left? 11.5. RECURRENCE OF BROWNIAN MOTION 11.6. THE POISSON EQUATION 11.7. ANALYSIS OF A SINGULAR BOUNDARY POINT 11.8. DISCOUNTING AND THE FEYNMAN-KAC FORMULA 11.8.1. Pricing of Bonds 11.8.2. Darwinian Fitness and Killing 11.8.3. Cumulated Rewards 11.8.4. Vertical Motion of Zooplankton 11.9. CONCLUSION 11.10. NOTES AND REFERENCES 11.11. EXERCISES CHAPTER 12: Stochastic Stability Theory 12.1. THE STABILITY PROBLEM 12.2. THE SENSITIVITY EQUATIONS 12.3. STOCHASTIC LYAPUNOV EXPONENTS 12.3.1. Lyapunov Exponent for a Particle in a Potential 12.4. EXTREMA OF THE STATIONARY DISTRIBUTION 12.5. A WORKED EXAMPLE: A STOCHASTIC PREDATOR-PREY MODEL 12.6. GEOMETRIC BROWNIAN MOTION REVISITED 12.7. STOCHASTIC LYAPUNOV FUNCTIONS 12.8. STABILITY IN MEAN SQUARE 12.9. STOCHASTIC BOUNDEDNESS 12.10. CONCLUSION 12.11. NOTES AND REFERENCES 12.12. EXERCISES CHAPTER 13: Dynamic Optimization 13.1. MARKOV DECISION PROBLEMS 13.2. CONTROLLED DIFFUSIONS AND PERFORMANCE OBJECTIVES 13.3. VERIFICATION AND THE HAMILTON-JACOBI-BELLMAN EQUATION 13.4. PORTFOLIO SELECTION 13.5. MULTIVARIATE LINEAR-QUADRATIC CONTROL 13.6. STEADY-STATE CONTROL PROBLEMS 13.6.1. Stationary LQR Control 13.7. DESIGNING AN AUTOPILOT 13.8. DESIGNING A FISHERIES MANAGEMENT SYSTEM 13.9. A FISHERIES MANAGEMENT PROBLEM IN 2D 13.10. OPTIMAL DIEL VERTICAL MIGRATIONS 13.11. CONCLUSION 13.12. NOTES AND REFERENCES 13.12.1. Control as PDE-Constrained Optimization 13.12.2. Numerical Analysis of the HJB Equation 13.13. EXERCISES CHAPTER 14: Perspectives Bibliography Index