Henderson D., Plaschko P. Stochastic Differential Equations in Science And Engineering

New Jersey: World Scientific, 2006. — 240 p. — ISBN: 981-256-296-6.Traditionally, non-quantum physics has been concerned with deterministic equations where the dynamics of the system are completely determined by initial conditions. A century ago the discovery of Brownian motion showed that nature need not be deterministic. However, it is only recently that there has been broad interest in nondeterministic and even chaotic systems, not only in physics but in ecology and economics. On a short term basis, the stock market is nondeterministic and often chaotic. Despite its significance, there are few books available that introduce the reader to modern ideas in stochastic systems. This book provides an introduction to this increasingly important field and includes a number of interesting applications.Stochastic Variables and Stochastic Processes.
Probability Theory.
Averages.
Stochastic Processes, the Kolmogorov Criterion and Martingales.
The Gaussian Distribution and Limit Theorems.
The central limit theorem.
Transformation of Stochastic Variables.
The Markov Property.
Stationary Markov processes.
The Brownian Motion.
Stochastic Integrals.
The Ito Formula.
Appendix: Poisson Processes.
Stochastic Differential Equations.
One-Dimensional Equations.
Growth of populations.
Stratonovich equations.
The problem of Ornstein-Uhlenbeck and the Maxwell distribution.
The reduction method.
Verification of solutions.
White and Colored Noise, Spectra.
The Stochastic Pendulum.
Stochastic excitation.
Stochastic damping (β = γ = 0; α ≠ 0).
The General Linear SDE.
A Class of Nonlinear SDE.
Existence and Uniqueness of Solutions.
The Fokker-Planck Equation.
The Master Equation.
The Derivation of the Fokker-Planck Equation.
The Relation Between the Fokker-Planck Equation and Ordinary SDE's.
Solutions to the Fokker-Planck Equation.
Lyapunov Exponents and Stability.
First order SDE's.
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.
Advanced Topics.
Stochastic Partial Differential Equations.
A deterministic one-dimensional wave equation.
Stochastic initial conditions.Mathematical methods.
Examples of exactly soluble problems.
Probability laws and moments of the eigenvalues.The Black-Scholes market.
Numerical Solutions of Ordinary Stochastic Differential Equations.
Random Numbers Generators and Applications.
Testing of random numbers.
The Convergence of Stochastic Sequences.
The Monte Carlo Integration.
The Brownian Motion and Simple Algorithms for SDE's.
The Ito-Taylor Expansion of the Solution of a 1D SDE.
Modified 1D Milstein Schemes.
The Ito-Taylor Expansion for N-dimensional SDE's.
Higher Order Approximations.
Strong and Weak Approximations and the Order of the Approximation.Fortran Programs.