Random Evolutions and Their Applications

The main purpose of this handbook is to summarize and to put in order the ideas, methods, results and literature on the theory of random evolutions and their applications to the evolutionary stochastic systems in random media, and also to present some new trends in the theory of random evolutions and their applications. In physical language, a random evolution ( RE ) is a model for a dynamical sys tem whose state of evolution is subject to random variations. Such systems arise in all branches of science. For example, random Hamiltonian and Schrodinger equations with random potential in quantum mechanics, Maxwell's equation with a random refractive index in electrodynamics, transport equations associated with the trajec tory of a particle whose speed and direction change at random, etc. There are the examples of a single abstract situation in which an evolving system changes its "mode of evolution" or "law of motion" because of random changes of the "environment" or in a "medium". So, in mathematical language, a RE is a solution of stochastic operator integral equations in a Banach space. The operator coefficients of such equations depend on random parameters. Of course, in such generality , our equation includes any homogeneous linear evolving system. Particular examples of such equations were studied in physical applications many years ago. A general mathematical theory of such equations has been developed since 1969, the Theory of Random Evolutions.

Inhalt
Definition and Classification of Semi-Markov Random Evolutions.- ESS in Semi-Markov Random Media.- Martingale Methods in RE.- Organization of the Handbook.- Structure of the Handbook.- Historical and Bibliographical Remarks to the Introduction.- 1 Multiplicative Operator Functionals.- 1.1 Semigroups of Operators.- 1.2 Additive and Multiplicative Functionals.- 1.3 Multiplicative Operator Functionals.- 1.4 Representations of MOF.- 1.5 Dual MOF.- 1.6 MOF Underlying Superprocesses.- 1.7 Stochastic Semigroups.- 1.8 Construction of a Markov Process by Multiplicative Functionals.- 1.9 Additive Operator Functionals.- 1.10 MOF on a Finite Markov Chain.- 2 Random Evolutions.- 2.1 Definition and Classification of Random Evolutions.- 2.2 Models of Random Evolutions.- 2.3 Evolutionary Equations.- 2.4 Martingale Methods in Random Evolutions.- 2.5 The Analogue of Dynkin's Formula for MOF and Random Evolutions.- 2.6 Boundary Value Problems for MOF and RE.- 2.7 Stability of Random Evolutions.- 2.8 Control of Random Evolution Historical and Bibliographical Remarks to Chapter 2.- 3 Limit Theorems for Random Evolutions.- 3.1 Limit theorems for models of random evolutions.- 3.2 Weak Convergence of Random Evolutions.- 3.3 Averaging of SMRE in a Series Scheme.- 3.4 Diffusion Approximation of SMRE in a Series Scheme.- 3.5 Averaging of SMRE in Reducible Phase Space. Merged RE.- 3.6 Diffusion Approximation of SMRE in a Reducible Phase Space.- 3.7 Normal deviations of SMRE.- 3.8 Rates of Convergence in the Limit Theorems for SMRE.- 3.9 Ergodic Theorem for MOF on a Markov Chain.- 4 Applications of Evolutionary Stochastic Systems.- 4.1 Random Evolutions as an Evolutionary Stochastic Systems in Random Media.- 4.2 Averaging and Merging of Evolutionary Stochastic Systems.- 4.3 DiffusionApproximation of Evolutionary Stochastic Systems.- 4.4 Rates of Convergence in the Limit Theorems for Stochastic Systems.- 4.5 Normal Deviations of Stochastic Systems.- 4.6 Stability of Evolutionary Stochastic Systems.- 4.7 Control of Evolutionary Stochastic Systems.- 5 New Trends in Random Evolutions.- 5.1 The Existence of the Wiener Measure and Related Stochastic Equations.- 5.2 Stochastic Integrals over Martingale Measures.- 5.3 Stochastic Integral Equations over Martingale Measures.- 5.4 Martingale Problems Connected with Stochastic Equations over Martingale Measures.- 5.5 Stochastic Integral Equation for Limiting Random Evolutions.- 5.6 Evolutionary Operator Equations Driven by the Wiener Martingale Measure.- 5.7 Random Evolutions in Financial Mathematics: Hedging of Options.