Solution Of Martingale Problem Research Articles

Statistical Causality, Extremal Measures and Weak Solutions of Stochastic Differential Equations with Driving Semimartingales

In this paper we consider different concepts of causality in filtered probability spaces. Especially, we consider a generalization of a causality relationship “G is a cause of J within H ” which was first given by Mykland (1986) and which is based on Granger’s definition of causality (Granger, Econometrica 37:424–438, 1969). Then we apply this concept on weak solutions of stochastic differential equations with driving semimartingales. We also show that the given causality concept is closely connected to the concept of extremality of measures and links Granger’s causality with the concept of adapted distribution. Finally, the concept of causality is applied on solution of martingale problem.

QUANTUM TRAJECTORIES IN RANDOM ENVIRONMENT: THE STATISTICAL MODEL FOR A HEAT BATH

In this article, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in \cite{A1P1}, where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

On statistical manifolds of solutions of martingale problems

On statistical manifolds of solutions of martingale problems

On statistical manifolds of solutions of martingale problems

We consider geometrically regular statistical models defined by local densities of probability measures corresponding to discrete or continuous time Markov processes and smoothly depending on a finite dimensional parameter. Evolution equations are derived in terms of the generators of the underlying Markov additive processes for the elements of the related Fisher information matrix and the skewness tensor defining the Riemannian metric and the Amari-Chentsov's affine α-connections as functions of time and starting points of Markov processes.

Convergence of Markov chains to solution of martingale problem

Convergence of Markov chains to solution of martingale problem