Local Relaxation and Collective Stochastic Dynamics

Damping and thermal fluctuations have been introduced to collective normal modes of a magnetic system in recent modeling of dynamic thermal magnetization processes. The connection between this collective stochastic dynamics and physical local relaxation processes is investigated here. A system of two coupled magnetic grains embedded in two separate oscillating thermal baths is analyzed with no \QTR{it}{a priori} assumptions except that of a Markovian process. It is shown explicitly that by eliminating the oscillating thermal bath variables, collective stochastic dynamics occurs in the normal modes of the magnetic system. The grain interactions cause local relaxation to be felt by the collective system and the dynamic damping to reflect the system symmetry. This form of stochastic dynamics is in contrast to a common phenomenological approach where a thermal field is added independently to the dynamic equations of each discretized cell or interacting grain. The dependence of this collective stochastic dynamics on the coupling strength of the magnetic grains and the relative local damping is discussed

The stability of electricity prices: estimation and inference of the Lyapunov exponents

The aim of this paper is to illustrate how the stability of a stochastic dynamic system is measured using the Lyapunov exponents. Specifically, we use a feedforward neural network to estimate these exponents as well as asymptotic results for this estimator to test for unstable (chaotic) dynamics. The data set used is spot electricity prices from the Nordic power exchange market. Nord Pool, and the dynamic system that generates these prices appears to be chaotic in one case.feedforward neural network; Nord Pool; Lyapunov exponents; spot electricity prices; stochastic dynamic system

Identification of dynamic systems, theory and formulation

The problem of estimating parameters of dynamic systems is addressed in order to present the theoretical basis of system identification and parameter estimation in a manner that is complete and rigorous, yet understandable with minimal prerequisites. Maximum likelihood and related estimators are highlighted. The approach used requires familiarity with calculus, linear algebra, and probability, but does not require knowledge of stochastic processes or functional analysis. The treatment emphasizes unification of the various areas in estimation in dynamic systems is treated as a direct outgrowth of the static system theory. Topics covered include basic concepts and definitions; numerical optimization methods; probability; statistical estimators; estimation in static systems; stochastic processes; state estimation in dynamic systems; output error, filter error, and equation error methods of parameter estimation in dynamic systems, and the accuracy of the estimates

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

In this paper we deal with the problem of existence of a smooth solution of the Hamilton-Jacobi-Bellman-Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.Comment: To appear in "Stochastic

Subjective Equilibria under Beliefs of Exogenous Uncertainty

We present a subjective equilibrium notion (called "subjective equilibrium under beliefs of exogenous uncertainty (SEBEU)" for stochastic dynamic games in which each player chooses its decisions under the (incorrect) belief that a stochastic environment process driving the system is exogenous whereas in actuality this process is a solution of closed-loop dynamics affected by each individual player. Players observe past realizations of the environment variables and their local information. At equilibrium, if players are given the full distribution of the stochastic environment process as if it were an exogenous process, they would have no incentive to unilaterally deviate from their strategies. This notion thus generalizes what is known as the price-taking equilibrium in prior literature to a stochastic and dynamic setup. We establish existence of SEBEU, study various properties and present explicit solutions. We obtain the $\epsilon$-Nash equilibrium property of SEBEU when there are many players