Markov chains approximation of jump–diffusion stochastic master equations

Clément Pellegrini

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 924-948
  • ISSN: 0246-0203

Abstract

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Quantum trajectories are solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also called Belavkin equations or Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump–diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab.36 (2008) 2332–2353] and [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.

How to cite

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Pellegrini, Clément. "Markov chains approximation of jump–diffusion stochastic master equations." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 924-948. <http://eudml.org/doc/239389>.

@article{Pellegrini2010,
abstract = {Quantum trajectories are solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also called Belavkin equations or Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump–diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab.36 (2008) 2332–2353] and [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.},
author = {Pellegrini, Clément},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic master equations; quantum trajectory; Jump–diffusion stochastic differential equation; stochastic convergence; Markov generators; martingale problem; jump-diffusion stochastic differential equation; Martingale problem},
language = {eng},
number = {4},
pages = {924-948},
publisher = {Gauthier-Villars},
title = {Markov chains approximation of jump–diffusion stochastic master equations},
url = {http://eudml.org/doc/239389},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Pellegrini, Clément
TI - Markov chains approximation of jump–diffusion stochastic master equations
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 924
EP - 948
AB - Quantum trajectories are solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also called Belavkin equations or Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump–diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab.36 (2008) 2332–2353] and [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.
LA - eng
KW - stochastic master equations; quantum trajectory; Jump–diffusion stochastic differential equation; stochastic convergence; Markov generators; martingale problem; jump-diffusion stochastic differential equation; Martingale problem
UR - http://eudml.org/doc/239389
ER -

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