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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 described by the Markov renewal
processes or by the semi Markov processes. The local
characteristics of the system depend on the state of the ran dom
medium. At the same time, the evolution of the system does not
affect the medium. Hence, the semi-Markov random evolutions are
described by two processes, namely, by the switching Markov renewal
process, which describes the changes of the state of the external
random medium, and by the switched process, i.e., by the semigroup
of oper ators describing the evolution of the system in the
semi-Markov random medium.
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."
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 securities prices, which is used for the description of
the evolution of bonds and stocks prices and also for their
derivatives, such as options, futures, forward contracts, etc., it
is supposed that the dynamic of bonds and stocks prices are set by
a linear differential and linear stochastic differential equations,
respectively, with interest rate, appreciation rate and volatility
such that they are predictable processes. Also, in the Arrow-Debreu
economy, the securities prices which support a Radner dynamic
equilibrium are a combination of an Ito process and a random point
process, with the all coefficients and jumps being predictable
processes."
The book is devoted to the study of limit theorems and stability of
evolving biologieal systems of "particles" in random environment.
Here the term "particle" is used broadly to include moleculas in
the infected individuals considered in epidemie models, species in
logistie growth models, age classes of population in demographics
models, to name a few. The evolution of these biological systems is
usually described by difference or differential equations in a
given space X of the following type and dxt/dt = g(Xt, y), here,
the vector x describes the state of the considered system, 9
specifies how the system's states are evolved in time (discrete or
continuous), and the parameter y describes the change ofthe
environment. For example, in the discrete-time logistic growth
model or the continuous-time logistic growth model dNt/dt =
r(y)Nt(l-Nt/K(y)), N or Nt is the population of the species at time
n or t, r(y) is the per capita n birth rate, and K(y) is the
carrying capacity of the environment, we naturally have X = R, X ==
Nn(X == Nt), g(x, y) = r(y)x(l-xl K(y)) , xE X. Note that n t for a
predator-prey model and for some epidemie models, we will have that
X = 2 3 R and X = R , respectively. In th case of logistic growth
models, parameters r(y) and K(y) normaIly depend on some random
variable y.
The book is devoted to the study of limit theorems and stability of
evolving biologieal systems of "particles" in random environment.
Here the term "particle" is used broadly to include moleculas in
the infected individuals considered in epidemie models, species in
logistie growth models, age classes of population in demographics
models, to name a few. The evolution of these biological systems is
usually described by difference or differential equations in a
given space X of the following type and dxt/dt = g(Xt, y), here,
the vector x describes the state of the considered system, 9
specifies how the system's states are evolved in time (discrete or
continuous), and the parameter y describes the change ofthe
environment. For example, in the discrete-time logistic growth
model or the continuous-time logistic growth model dNt/dt =
r(y)Nt(l-Nt/K(y)), N or Nt is the population of the species at time
n or t, r(y) is the per capita n birth rate, and K(y) is the
carrying capacity of the environment, we naturally have X = R, X ==
Nn(X == Nt), g(x, y) = r(y)x(l-xl K(y)) , xE X. Note that n t for a
predator-prey model and for some epidemie models, we will have that
X = 2 3 R and X = R , respectively. In th case of logistic growth
models, parameters r(y) and K(y) normaIly depend on some random
variable y.
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 securities prices, which is used for the description of
the evolution of bonds and stocks prices and also for their
derivatives, such as options, futures, forward contracts, etc., it
is supposed that the dynamic of bonds and stocks prices are set by
a linear differential and linear stochastic differential equations,
respectively, with interest rate, appreciation rate and volatility
such that they are predictable processes. Also, in the Arrow-Debreu
economy, the securities prices which support a Radner dynamic
equilibrium are a combination of an Ito process and a random point
process, with the all coefficients and jumps being predictable
processes."
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."
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 described by the Markov renewal
processes or by the semi Markov processes. The local
characteristics of the system depend on the state of the ran dom
medium. At the same time, the evolution of the system does not
affect the medium. Hence, the semi-Markov random evolutions are
described by two processes, namely, by the switching Markov renewal
process, which describes the changes of the state of the external
random medium, and by the switched process, i.e., by the semigroup
of oper ators describing the evolution of the system in the
semi-Markov random medium."
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