Grigoriu M. Stochastic Systems: Uncertainty Quantification and Propagation

Springer, 2012. — 534 p. — (Springer Series in Reliability Engineering). — ISBN: 1447123271.Equations are used in applied sciences and engineering to characterize the state of mechanical, economics, physical, environmental, and other systems. For a long time it has been assumed that the coefficients of, the input to, and the end conditions for these equations are deterministic and perfectly known. Yet, owing to inherent variability and/or incomplete knowledge, the entries of most state equations are uncertain. Equations with deterministic, perfectly known entries are called deterministic equations (DEs); those with random entries are called stochastic equations (SEs). Although DEs are useful in many cases, theymay provide limited information even on the trend of a system state. In contrast, SEs can capture both the trend and variability of a system state.Essentials of Probability Theory
Random Functions
Stochastic Integrals
Itô’s Formula and Applications
Probabilistic Models
Stochastic Ordinary Differential and Difference Equations
Stochastic Algebraic Equations
Stochastic Partial Differential Equations
Appendixes
Parametric Models, Quantizers, and Stochastic Reduced Order Models
A Primer on Functional