Stochastic Differential Equations for Science and Engineering

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