Optimal Multi-Modes Switching Problem in Infinite Horizon

This paper studies the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a finne analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. This problem is in relation with the valuation of firms in a financial market

Viscosity Solutions for a System of PDEs and Optimal Switching

In this paper, we study the $m$-states optimal switching problem in finite horizon, when the switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central evaluation in cases where such organizations or state grants or financial assistance to power plants that promotes green energy in their production activity or what uses less polluting modes in their production. We show existence for optimal strategy via a verification theorem then we show existence and uniqueness of the value processes by using an approximation scheme. In the markovian framework we show that the value processes can be characterized in terms of deterministic continuous functions of the state of the process. Those latter functions are the unique viscosity solutions for a system of $m$ variational partial differential inequalities with inter-connected obstacles.Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1102.1256, arXiv:0805.1306, arXiv:0904.0707, arXiv:1202.1108, and arXiv:0707.2663 and arXiv:1104.2689 by other authors. IMA Journal of Mathematical Control and Information (2016

The ``Mixed'' Green's Function Approach to Quantum Kinetics with Initial Correlations

A method for deriving quantum kinetic equations with initial correlations is developed on the basis of the nonequilibrium Green's function formalism. The method is applicable to a wide range of correlated initial states described by nonequilibrium statistical thermodynamics. Initial correlations and the real-time evolution are treated by a unified technique employing many-component ``mixed'' Green's functions. The Dyson equation for the mixed Green's function leads to a set of equations for real-time Green's functions and new (cross) components linking initial correlations with dynamical processes. These equations are used to formulate a generalized Kadanoff-Baym ansatz for correlated initial states. A non-Markovian short-time kinetic equation is derived within the T-matrix approximation for the self-energies. The properties of the memory kernels in this equation are considered in detail in Born approximation for the T-matrices. The kinetic equation is demonstrated to conserve the total energy of the system. An explicit expression for the time-dependent correlation energy is obtained.Comment: 50 pages, 3 figure

Theory of non-Markovian Stochastic Resonance

We consider a two-state model of non-Markovian stochastic resonance (SR) within the framework of the theory of renewal processes. Residence time intervals are assumed to be mutually independent and characterized by some arbitrary non-exponential residence time distributions which are modulated in time by an externally applied signal. Making use of a stochastic path integral approach we obtain general integral equations governing the evolution of conditional probabilities in the presence of an input signal. These novel equations generalize earlier integral renewal equations by Cox and others to the case of driving-induced non-stationarity. On the basis of these new equations a response theory of two state renewal processes is formulated beyond the linear response approximation. Moreover, a general expression for the linear response function is derived. The connection of the developed approach with the phenomenological theory of linear response for manifest non-Markovian SR put forward in [ I. Goychuk and P. Hanggi, Phys. Rev. Lett. 91, 070601 (2003)] is clarified and its range of validity is scrutinized. The novel theory is then applied to SR in symmetric non-Markovian systems and to the class of single ion channels possessing a fractal kinetics

Reduced hierarchy equations of motion approach with Drude plus Brownian spectral distribution: Probing electron transfer processes by means of two- dimensionalcorrelation spectroscopy

We theoretically investigate an electron transfer (ET) process in a dissipative environment by means of two-dimensional (2D) correlation spectroscopy. We extend the reduced hierarchy equations of motion approach to include both overdamped Drude and underdamped Brownian modes. While the overdamped mode describes the inhomogeneity of a system in the slow modulation limit, the underdamped mode expresses the primary vibrational mode coupled with the electronic states. We outline a procedure for calculating 2D correlation spectrum that incorporates the ET processes. The present approach has the capability of dealing with system-bath coherence under an external perturbation, which is important to calculate nonlinear response functions for non-Markovian noise. The calculated 2D spectrum exhibits the effects of the ET processes through the presence of ET transition peaks along the $\Omega_1$ axis, as well as the decay of echo signals.Comment: 28 pages, 8 figures; J. Chem. Phys. 137 (2012

On Markovian Cocycle Perturbations in Classical and Quantum Probability

We introduce Markovian cocycle perturbations of the groups of transformations associated with the classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. It is namely the definition one needs because the Markovian perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed in the sum of a part associated with deterministic stochastic processes lying in the past and a part associated with the noise isomorphic to the initial one. This decomposition allows to obtain some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increaments it is constructed the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations. Using this model we construct Markovian cocyclies transformating the Gaussian state $\rho$ to the Gaussian states equivalent to $\rho$.Comment: 27 page

Concepts of quantum non-Markovianity: a hierarchy

Markovian approximation is a widely-employed idea in descriptions of the dynamics of open quantum systems (OQSs). Although it is usually claimed to be a concept inspired by classical Markovianity, the term quantum Markovianity is used inconsistently and often unrigorously in the literature. In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial problem of characterizing quantum non-Markovianity. Rather than proposing a single definition of quantum Markovianity, we study a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, such as quantum white noise, factorization approximation, divisibility, Lindblad master equation, etc.. Others are first proposed in this report, including those we call past-future independence, no (quantum) information backflow, and composability. All of these concepts are defined under a unified framework, which allows us to rigorously build hierarchy relations among them. With various examples, we argue that the current most often used definitions of quantum Markovianity in the literature do not fully capture the memoryless property of OQSs. In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity.Comment: Clarifications and references added; discussion of the related classical hierarchy significantly improved. To appear in Physics Report

A genetic approach to Markovian characterisation of H.264 scalable video

We propose an algorithm for multivariate Markovian characterisation of H.264/SVC scalable video traces at the sub-GoP (Group of Pictures) level. A genetic algorithm yields Markov models with limited state space that accurately capture temporal and inter-layer correlation. Key to our approach is the covariance-based fitness function. In comparison with the classical Expectation Maximisation algorithm, ours is capable of matching the second order statistics more accurately at the cost of less accuracy in matching the histograms of the trace. Moreover, a simulation study shows that our approach outperforms Expectation Maximisation in predicting performance of video streaming in various networking scenarios

Markovian Characterisation of H.264/SVC scalable video

In this paper, a multivariate Markovian traffic: model is proposed to characterise H.264/SVC scalable video traces. Parametrisation by a genetic algorithm results in models with a limited state space which accurately capture. both the temporal and the inter-layer correlation of the traces. A simulation study further shows that the model is capable of predicting performance of video streaming in various networking scenarios