The book is devoted to the new trends in random evolutions and their various applications to stochastic evolutionary sytems (SES). Such new developments as the analogue of Dynkin's formulae, boundary value problems, stochastic stability and optimal control of random evolutions, stochastic evolutionary equations driven by martingale measures are considered. The book also contains such new trends in applied probability as stochastic models of financial and insurance mathematics in an incomplete market. In the famous classical financial mathematics Black-Scholes model of a (B, S) market for...

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...

The evolution of systems in random media is a broad and fruitful field for the applica tions of different mathematical methods and theories. This evolution can be character ized by a semigroup property. In the abstract form, this property is given by a semigroup of operators in a normed vector (Banach) space. In the practically boundless variety of mathematical models of the evolutionary systems, we have chosen the semi-Markov ran dom evolutions as an object of our consideration. The definition of the evolutions of this type is based on rather simple initial assumptions. The random medium is...