6,444 research outputs found
On the form of the large deviation rate function for the empirical measures of weakly interacting systems
A basic result of large deviations theory is Sanov's theorem, which states
that the sequence of empirical measures of independent and identically
distributed samples satisfies the large deviation principle with rate function
given by relative entropy with respect to the common distribution. Large
deviation principles for the empirical measures are also known to hold for
broad classes of weakly interacting systems. When the interaction through the
empirical measure corresponds to an absolutely continuous change of measure,
the rate function can be expressed as relative entropy of a distribution with
respect to the law of the McKean-Vlasov limit with measure-variable frozen at
that distribution. We discuss situations, beyond that of tilted distributions,
in which a large deviation principle holds with rate function in relative
entropy form.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ540 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Discretisation of Stochastic Control Problems for Continuous Time Dynamics with Delay
As a main step in the numerical solution of control problems in continuous time, the controlled process is approximated by sequences of controlled Markov chains, thus discretizing time and space. A new feature in this context is to allow for delay in the dynamics. The existence of an optimal strategy with respect to the cost functional can be guaranteed in the class of relaxed controls. Weak convergence of the approximating extended Markov chains to the original process together with convergence of the associated optimal strategies is established.Markov, Markov chain, time dynamics, stochastic control problem
On large deviations for small noise It\^o processes
The large deviation principle in the small noise limit is derived for
solutions of possibly degenerate It\^o stochastic differential equations with
predictable coefficients, which may depend also on the large deviation
parameter. The result is established under mild assumptions using the
Dupuis-Ellis weak convergence approach. Applications to certain systems with
memory and to positive diffusions with square-root-like dispersion coefficient
are included.Comment: 30 page
Continuous time mean-variance portfolio optimization through the mean field approach
A simple mean-variance portfolio optimization problem in continuous time is solved using the mean field approach. In this approach, the original optimal control problem, which is time inconsistent, is viewed as the McKean\u2013Vlasov limit of a family of controlled many-component weakly interacting systems. The prelimit problems are solved by dynamic programming, and the solution to the original problem is obtained by passage to the limit
A two state model for noise-induced resonance in bistable systems with delay
The subject of the present paper is a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one dimensional parametrized stochastic differential equation with point delay, basic properties whereof we check. With a view to capturing the effective dynamics and, in particular, the resonance-like behaviour of the reference model we construct a simplified or reduced model, the two state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn can be used to characterize the phenomenon of noise-induced resonance. Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two state model and the reference model. The resonance characteristics developed for the reduced model can thus be applied to the original model.logit model, utility maximization nested logit, non-normalized nested logit, simulation study
On Iterated Dominance, Matrix Elimination, and Matched Paths
We study computational problems arising from the iterated removal of weakly
dominated actions in anonymous games. Our main result shows that it is
NP-complete to decide whether an anonymous game with three actions can be
solved via iterated weak dominance. The two-action case can be reformulated as
a natural elimination problem on a matrix, the complexity of which turns out to
be surprisingly difficult to characterize and ultimately remains open. We
however establish connections to a matching problem along paths in a directed
graph, which is computationally hard in general but can also be used to
identify tractable cases of matrix elimination. We finally identify different
classes of anonymous games where iterated dominance is in P and NP-complete,
respectively.Comment: 12 pages, 3 figures, 27th International Symposium on Theoretical
Aspects of Computer Science (STACS
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