3,882 research outputs found
Switching Diffusions: Applications To Ecological Models, And Numerical Methods For Games In Insurance
Recently, a class of dynamic systems called ``hybrid systems containing both continuous dynamics and discrete events has been adapted to treat a wide variety of situations arising in many real-world situations. Motivated by such development, this dissertation is devoted to the study of dynamical systems involving a Markov chain as the randomly switching process. The systems studied include hybrid competitive Lotka-Volterra ecosystems and non-zero-sum stochastic differential games between two insurance companies with regime-switching.
The first part is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective is to reduce the computational complexity by using the two-time-scale formulation. Because the existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this work that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex original system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods.
The second part develops an approximation procedure for a class of non-zero-sum stochastic differential games for investment and reinsurance between two insurance companies. Both proportional reinsurance and excess-of-loss reinsurance policies are considered. We develop numerical algorithms to obtain the approximation to the Nash equilibrium by adopting the Markov chain approximation methodology. We establish the convergence of the approximation sequences and the approximation to the value functions. Numerical examples are presented to illustrate the applicability of the algorithms
Mean Field Control for Efficient Mixing of Energy Loads
We pose an engineering challenge of controlling an Ensemble of Energy Devices
via coordinated, implementation-light and randomized on/off switching as a
problem in Non-Equilibrium Statistical Mechanics. We show that Mean Field
Control} with nonlinear feedback on the cumulative consumption, assumed
available to the aggregator via direct physical measurements of the energy
flow, allows the ensemble to recover from its use in the Demand Response
regime, i.e. transition to a statistical steady state, significantly faster
than in the case of the fixed feedback. Moreover when the nonlinearity is
sufficiently strong, one observes the phenomenon of "super-relaxation" -- where
the total instantaneous energy consumption of the ensemble transitions to the
steady state much faster than the underlying probability distribution of the
devices over their state space, while also leaving almost no devices outside of
the comfort zone.Comment: 7 pages, 5 figure
Pathogen evolution in switching environments: a hybrid dynamical system approach
We propose a hybrid dynamical system approach to model the evolution of a
pathogen that experiences different selective pressures according to a
stochastic process. In every environment, the evolution of the pathogen is
described by a version of the Fisher-Haldane-Wright equation while the
switching between environments follows a Markov jump process. We investigate
how the qualitative behavior of a simple single-host deterministic system
changes when the stochastic switching process is added. In particular, we study
the stability in probability of monomorphic equilibria. We prove that in a
"constantly" fluctuating environment, the genotype with the highest mean
fitness is asymptotically stable in probability. However, if the probability of
host switching depends on the genotype composition of the population,
polymorphism can be stably maintained.
This is a corrected version of the paper that appeared in Mathematical
Biosciences 240 (2012), p. 70-75. A corrigendum has appeared in the same
journal.Comment: 15 pages, 4 figure
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE
We aim to provide a Feynman-Kac type representation for
Hamilton-Jacobi-Bellman equation, in terms of forward backward stochastic
differential equation (FBSDE) with a simulatable forward process. For this
purpose, we introduce a class of BSDE where the jumps component of the solution
is subject to a partial nonpositive constraint. Existence and approximation of
a unique minimal solution is proved by a penalization method under mild
assumptions. We then show how minimal solution to this BSDE class provides a
new probabilistic representation for nonlinear integro-partial differential
equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a
regime switching forward SDE in a Markovian framework, and importantly we do
not make any ellipticity condition. Moreover, we state a dual formula of this
BSDE minimal solution involving equivalent change of probability measures. This
gives in particular an original representation for value functions of
stochastic control problems including controlled diffusion coefficient.Comment: Published at http://dx.doi.org/10.1214/14-AOP920 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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