CONTENTS -- Preface -- Introduction -- Glossary -- 1. Stochastic Variables and Stochastic Processes -- 1.1. Probability Theory -- 1.2. Averages -- 1.3. Stochastic Processes the Kolmogorov Criterion and Martingales -- 1.4. The Gaussian Distribution and Limit Theorems -- 1.4.1. The central limit theorem -- 1.4.2. The law of the iterated logarithm -- 1.5. Transformation of Stochastic Variables -- 1.6. The Markov Property -- 1.6.1. Stationary Markov processes -- 1.7. The Brownian Motion -- 1.8. Stochastic Integrals -- 1.9. The Ito Formula -- Appendix: Poisson Processes -- Exercises -- 2. Stochastic Differential Equations -- 2.1. One-Dimensional Equations -- 2.1.1. Growth of populations -- 2.1.2. Stratonovich equations -- 2.1.3. The problem of Ornstein-Uhlenbeck and the Maxwell distribution -- 2.1.4. The reduction method -- 2.1.5. Verification of solutions -- 2.2. White and Colored Noise Spectra -- 2.3. The Stochastic Pendulum -- 2.3.1. Stochastic excitation -- 2.3.2. Stochastic damping (B = r = 0 -- a # 0) -- 2.4. The General Linear SDE -- 2.5. A Class of Nonlinear SDE -- 2.6. Existence and Uniqueness of Solutions -- Exercises -- 3. The Fokker-Planck Equation -- 3.1. The Master Equation -- 3.2. The Derivation of the Fokker-Planck Equation -- 3.3. The Relation Between the Fokker-Planck Equation and Ordinary SDE's -- 3.4. Solutions to the Fokker-Planck Equation -- 3.5. Lyapunov Exponents and Stability -- 3.6. Stochastic Bifurcations -- 3.6.1. First order SDE's -- 3.6.2. Higher order SDE's -- Appendix A. Small Noise Intensities and the Influence of Randomness Limit Cycles -- Appendix B.1 The method of Lyapunov functions -- Appendix B.2 The method of linearization -- Exercises -- 4. Advanced Topics -- 4.1. Stochastic Partial Differential Equations -- 4.2. Stochastic Boundary and Initial Conditions.

4.2.1. A deterministic one-dimensional wave equation -- 4.2.2. Stochastic initial conditions -- 4.3. Stochastic Eigenvalue Equations -- 4.3.1. Introduction -- 4.3.2. Mathematical methods -- 4.3.3. Examples of exactly soluble problems -- 4.3.4. Probability laws and moments of the eigenvalues -- 4.4. Stochastic Economics -- 4.4.1. Introduction -- 4.4.2. The Black-Scholes market -- Exercises -- 5. Numerical Solutions of Ordinary Stochastic Differential Equations -- 5.1. Random Numbers Generators and Applications -- 5.1.1. Testing of random numbers -- 5.2. The Convergence of Stochastic Sequences -- 5.3. The Monte Carlo Integration -- 5.4. The Brownian Motion and Simple Algorithms for SDE's -- 5.5. The Ito-Taylor Expansion of the Solution of a 1D SDE -- 5.6. Modified 1D Milstein Schemes -- 5.7. The Ito-Taylor Expansion for N-dimensional SDE's -- 5.8. Higher Order Approximations -- 5.9. Strong and Weak Approximations and the Order of the Approximation -- Exercises -- References -- Fortran Programs -- Index.