determinacy indeterminacy and dynamic misspecification in linear rational expectations models abstract this paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations lre model may have a unique stable solution determinacy against the alternative of multiple stable solutions indeterminacy the testing problem is addressed by a misspecification type approach in which the overidentifying restrictions test obtained from the estimation of the system of euler equations of the lre model through the generalized method of moments is combined with a likelihood based test for the cross equation restrictions that the model places on its reduced form solution under determinacy the resulting test has no power against a particular class of indeterminate equilibria hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy on the other hand this test i circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy ii does not involve inequality parametric restrictions and hence the use of nonstandard inference iii is consistent against the dynamic misspecification of the lre model and iv is computationally simple monte carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples an empirical illustration focuses on the determinacy indeterminacy of a new keynesian monetary business cycle model of the us economy

hardy s inequality and the l1 norm of exponential sums in this paper we generalize hardy s inequality 3 for measures of analytic type and obtain a proof of the littlewood conjecture 4 for the l norm of exponential sums as a simple consequence let t be the circle group z the additive group of integers and m t the customary convolution algebra of borel measures on t for pe e m t and n g z put m e ino dp o denote by h t the classical space of all measures e e m t such that j n 0 for all n 0 and let c denote the complex numbers we now state hardy s inequality for measures of analytic type theorem 1 if m c h t then

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

quantum pontryagin principle under continuous measurements in this paper we develop the theory of the quantum pontryagin principle for continuous measurements and feedback the analysis is carried out under the assumption of compatible events in the output channel the plant is a quantum system which generally is in a mixed state coupled to a continuous measurement channel the pontryagin maximum principle is derived in both the schrodinger picture and heisenberg picture in particular in statistical moment coordinates to avoid solving stochastic equations we derive a linear quadratic gaussian scheme which is more suitable for control purposes finally we use the quantum harmonic oscillator as a concrete example to illustrate the performance of the controller

quantum measurement based feedback control a nonsmooth time delay control approach this paper addresses the problem of measurement based feedback control for quantum systems in which the time to compute the filter based control input is taken into account by considering the input delay it starts with a new lyapunov lasalle like theorem for delay dependent stochastic stability of a class of time delay stochastic nonlinear systems nonsmooth time delay control is then constructed to compensate for the control computation time which is known but arbitrarily long while globally stabilizing the quantum filters almost surely the nonsmooth property enables the control to deal with the symmetric topology of filter state space the effectiveness of the proposed control is illustrated through the global stabilization of the spin 1 2 systems simulation results are presented and discussed to show the effectiveness of the proposed control

large time behavior and convergence rate for quantum filters under standard non demolition conditions a quantum system mathcal s undergoing continuous time measurement is usually described by a jump diffusion stochastic differential equation such an equation is called a quantum filtering equation or quantum stochastic master equation and its solution is called a quantum filter or quantum trajectory this solution describes the evolution of the state of mathcal s in the context of quantum non demolition measurement we investigate the large time behavior of this solution it is rigorously shown that for large time this solution behaves as if a direct von neumann measurement has been performed at time 0 in particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate using the theory of girsanov transformation we obtain the explicit rate of convergence towards this random state the problem of state estimation used in experiment is also investigated

information geometry and local asymptotic normality for multi parameter estimation of quantum markov dynamics this paper deals with the problem of identifying and estimating dynamical parameters of continuous time markovian quantum open systems in the input output formalism first we characterise the space of identifiable parameters for ergodic dynamics assuming full access to the output state for arbitrarily long times and show that the equivalence classes of undistinguishable parameters are orbits of a lie group acting on the space of dynamical parameters second we define an information geometric structure on this space including a principal bundle given by the action of the group as well as a compatible connection and a riemannian metric based on the quantum fisher information of the output we compute the metric explicitly in terms of the markov covariance of certain fluctuation operators and relate it to the horizontal bundle of the connection third we show that the system output and reduced output state satisfy local asymptotic normality i e they can be approximated by a gaussian model consisting of coherent states of a multimode continuos variables system constructed from the markov covariance data we illustrate the result by working out the details of the information geometry of a physically relevant two level system

hardy s inequality and fractal measures abstract hardy s inequality and the subsequent improvement by mcgehee pigno and smith are generalized from the positive integers to sets of dimension 0 dimension 1 and in between the asymptotic estimate obtained for the fourier transform of fractal measures is much in the spirit of recent work by strichartz

the series product and its application to quantum feedforward and feedback networks the purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks these methods build on and generalize existing methods for series or cascade connections by allowing for more general interfaces and by introducing an efficient algebraic tool the series product we also introduce another product which we call the concatenation product that is useful for assembling and representing systems without necessarily having connections we show how the concatenation and series products can be used to describe feedforward and feedback networks a selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology

quantum optical realization of classical linear stochastic systems abstract the purpose of this paper is to show how a class of classical linear stochastic systems can be physically implemented using quantum optical components quantum optical systems typically have much higher bandwidth than electronic devices meaning faster response and processing times and hence have the potential for providing better performance than classical systems a procedure is provided for constructing the quantum optical realization the paper also describes the use of the quantum optical realization in a measurement feedback loop some examples are given to illustrate the application of the main results

nearly efficient estimation of time series models with predetermined but not exogenous instruments particularly under the assumption of rational expectations a model may have serially correlated errors and those errors may be uncorrelated with contemporaneous and lagged values of a predetermined instrument yet the instruments may not be strictly exogenous this paper proposes a method for transforming such a model to one without serial correlation while keeping the instrument predetermined standard theory of instrumental variables estimation then applies furthermore it turns out that for transformations of the class proposed asymptotic distribution theory is the same whether the serial correlation properties of the errors are known a priori or estimated as the number of lagged values of the predetermined variables used as instruments increases the asymptotic variance of the standard instrumental variables estimator applied to the transformed model approaches that of the optimal estimator proposed by hansen and sargent 8 in a number of recently developed macroeconomic models behavioral equations arise in which error terms can be asserted on the basis of economic arguments to be uncorrelated with some set of instrumental variables at a certain set of dates but not to be uncorrelated with the instruments at all dates examples of such

full versus limited information estimation of a rational expectations model some numerical comparisons abstract this paper compares numerically the asymptotic distributions of parameter estimates and test statistics associated with two estimation techniques a a limited information one which uses instrumental variables to estimate a single equation hansen and singleton 1982 and b a full information one which uses a procedure asymptotically equivalent to maximum likelihood to simultaneously estimate multiple equations hansen and sargent 1980 the paper compares the two with respect to both 1 asymptotic efficiency under the null hypothesis of no misspecification and 2 asymptotic bias and power in the presence of certain local alternatives it is found that 1 full information standard errors are only moderately smaller than limited information standard errors and 2 when the model is misspecified full information tests tend to be more powerful and its parameter estimates tend to be more biased this suggests that at least in the model considered here the gains from the use of the less robust and computationally more complex full information technique are not particularly large

local asymptotic specification error analysis an approximation to the inconsistency introduced by imposing an incorrect restriction on a parametric model is given the approximation can be applied to estimators generated by optimizing any objective function satisfying certain regularity conditions examples given include analysis of misspecification in discrete choice and time series models estimated by maximum likelihood and in a nonlinear regression model specification error analysis in the linear regression model has been studied by theil 1 who gives formulas for e g the effect of leaving out relevant variables on the expected values of the estimators of the coefficients of the included variables in this paper we suggest analogous formulas for estimators obtained by optimizing an objective function subject to restrictions we have in mind maximizing 1 n x loglikelihood and will usually use this terminology we consider the effect on the limit of the restricted estimator of a small violation of the restrictions in the linear regression case our formula coincides with that given by theil in order to keep our results widely applicable and to avoid a mass of unnecessary detail we make assumptions on the asymptotic behavior of the loglikelihood function itself rather than on the data generating process per se many alternative sets of assumptions on the data densities can lead to the behavior we require of the loglikelihood functions these will not be pursued here the interested reader is referred to e g white 12 for the case of independent observations and kohn 8 for the time series case 1 general formulas the general approach we take is based on a linear approximation to the likelihood function at the maximum likelihood estimator it is in this sense that our analysis is local for some models the local and global specification error results coincide a well known case is the effect of omitted regressors in the linear regression model essentially the only cases involve linearity although often there is agreement regarding the signs of the inconsistency we show below that the local and global results even fail to coincide in the case of misspecified ar processes generally however the global results are unknown 2 taylor expansions are typically used together with assumptions on the data generating process to obtain the asymptotic distribution of the maximum likelihood estimator cramer 3 in this paper we will not concern ourselves with asymptotic distributions of vn normed mle s since these have been worked

markov dilations on w algebras abstract in this paper a systematic study of markov dilations is begun for completely positive operators on w algebras which leave a faithful normal state invariant it is shown that a minimal markov dilation preserves important properties of the underlying completely positive operator afterwards some results are proved concerning the construction of dilations which lead to markov dilations for large classes of operators finally some of the ideas developed here are applied to the study of a simple example over the 2 2 matrices

self similar measures and their fourier transforms ii a self similar measure on r n was defined by hutchinson to be a probability measure satisfying formule here s j x j r j x b j is a contractive similarity 0 j 1 r j orthogonal and the weights a j satisfy 0 a j 1 j 1 m a j 1 by analogy we define a self similar distribution by the same identity but allowing the weights a j to be arbitrary complex numbers we give necessary and sufficient conditions for the existence of a solution to among distributions of compact support and show that the space of such solutions is always finite dimensional if f denotes the fourier transformation of a self similar distribution of compact support let formule where is defined by the equation j 1 m j a j 2 1

feedback control of quantum state reduction feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement we formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter we explore the use of stochastic lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability

weighted norm estimates for the fourier transform with a pair of weights we prove weighted norm inequalities of the form iifiilq ciifiihp 0 p q 0 for the fourier transform on rn for some weight functions v the hardy space hp on the right can be replaced by l the proof depends on making an atomic decomposition of f and using cancellation properties of the atoms

singular perturbation approximations for a class of linear quantum systems this technical note considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics the technical note presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction

coherent observers for linear quantum stochastic systems the theory of observers is a basic part of classical linear system theory the purpose of this paper is to develop a theory of coherent observers for linear quantum systems we provide a class of coherent quantum observers which track the observables of a linear quantum stochastic system in the sense of mean values independent of any additional quantum noise in the observer we prove that there always exists such a coherent quantum observer described by quantum stochastic differential equations in the heisenberg picture and show how it can be designed to be consistent with the laws of quantum mechanics we also find a lower bound for the mean squared estimation error due to the uncertainty principle in addition we explore the quantum correlations between a linear quantum plant and the corresponding coherent observer it is shown that considering a joint plant observer gaussian quantum system entanglement can be generated under the condition that appropriate coefficients of the coherent quantum observer are chosen and this issue is illustrated in an example these results pave the way towards observer based quantum control

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

an introduction to quantum filtering this paper provides an introduction to quantum filtering theory an introduction to quantum probability theory is given focusing on the spectral theorem and the conditional expectation as a least squares estimate and culminating in the construction of wiener and poisson processes on the fock space we describe the quantum ito calculus and its use in the modeling of physical systems we use both reference probability and innovations methods to obtain quantum filtering equations for system probe models from quantum optics

fidelity is a sub martingale for discrete time quantum filters fidelity is known to increase through any kraus map the fidelity between two density matrices is less than the fidelity between their images via a kraus map we prove here that in average fidelity is also increasing for discrete time quantum filters attached to an arbitrary kraus map fidelity between the density matrix of the underlying markov chain and the density matrix of the associated quantum filter is a sub martingale this result is not restricted to pure states it also holds true for mixed states

coupled ito equations of continuous quantum state measurement and estimation we discuss a nonlinear stochastic master equation that governs the time evolution of the estimated quantum state its differential evolution corresponds to the infinitesimal updates that depend on the time continuous measurement of the true quantum state the new stochastic master equation couples to the two standard stochastic differential equations of time continuous quantum measurement for the first time we can prove that the calculated estimate almost always converges to the true state also at low efficiency measurements we show that our single state theory can be adapted to weak continuous ensemble measurements as well

quasi free quantum stochastic integrals for the car and ccr abstract a theory of quantum martingales and quantum stochastic integrals in quasi free representations of the car and ccr is presented for the car the results generalize some of those developed in barnett streater and wilde j funct anal 48 1982 172 212 j london math soc 27 1983 373 384 and for the ccr the results contain the standard ito theory of stochastic integration with respect to brownian motion as a special case