Stochastic Differential Equations and Applications

PrefaceGeneral NotationContents of Volume 21 Stochastic Processes 1. The Kolmogorov Construction of a Stochastic Process 2. Separable and Continuous Processes 3. Martingales and Stopping Times Problems2 Markov Processes 1. Construction of Markov Processes 2. The Feller and the Strong Markov Properties 3. Time-Homogeneous Markov Processes Problems3 Brownian Motion 1. Existence of Continuous Brownian Motion 2. Nondifferentiability of Brownian Motion 3. Limit Theorems 4. Brownian Motion After a Stopping Time 5. Martingales and Brownian Motion 6. Brownian Motion in n Dimensions Problems4 The Stochastic Integral 1. Approximation of Functions by Step Functions 2. Definition of the Stochastic Integral 3. The Indefinite Integral 4. Stochastic Integrals with Stopping Time 5. Itôs Formula 6. Applications of Itôs Formula 7. Stochastic Integrals and Differentials in n Dimensions Problems5 Stochastic Differential Equations 1. Existence and Uniqueness 2. Stronger Uniqueness and Existence Theorems 3. The Solution of a Stochastic Differential System as a Markov Process 4. Diffusion Processes 5. Equations Depending on a Parameter 6. The Kolmogorov Equation Problems6 Elliptic and Parabolic Partial Differential Equations and Their Relations to Stochastic Differential Equations 1. Square Root of a Nonnegative Definite Matrix 2. The Maximum Principle for Elliptic Equations 3. The Maximum Principle for Parabolic Equations 4. The Cauchy Problem and Fundamental Solutions for Parabolic Equations 5. Stochastic Representation of Solutions of Partial Differential Equations Problems7 The Cameron-Martin-Girsanov Theorem 1. A Class of Absolutely Continuous Probabilities 2. Transformation of Brownian Motion 3. Girsanov's Formula Problems8 Asymptotic Estimates for Solutions 1. Unboundedness of Solutions 2. Auxiliary Estimates 3. Asymptotic Estimates 4. Applications of the Asymptotic Estimates 5. The One-Dimensional Case 6. Counterexample Problems9 Recurrent and Transient Solutions 1. Transient Solutions 2. Recurrent Solutions 3. Rate of Wandering Out to Infinity 4. Obstacles 5. Transient Solutions for Degenerate Diffusion 6. Recurrent Solutions for Degenerate Diffusion 7. The One-Dimensional Case ProblemsBibliographical RemarksReferencesIndex