adaptive control of linear systems with markov perturbations the stochastic model considered is a linear jump diffusion process x for which the coefficients and the jump processes depend on a markov chain z with finite state space first we study the optimal filtering and control problem for these systems with non gaussian initial conditions given noisy observations of the state x and perfect measurements of z we derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter we study a quadratic control problem and show that the separation principle holds next we investigate an adaptive control problem for a state process x defined by a linear diffusion for which the coefficients depend on a markov chain the processes x and z being observed in independent white noises suboptimal estimates for the process x z and approximate control law are investigated for a large class of probability distributions of the initial state asymptotic properties of these filters and this control law are obtained upper bounds for the corresponding error are given

adaptive estimation in a random coefficient autoregressive model this paper proves the local asymptotic normality of a stationary and ergodic first order random coefficient autoregressive model in a semiparametric setting this result is used to show that stein s necessary condition for adaptive estimation of the mean of the random coefficient is satisfied if the distributions of the innovations and the errors in the random coefficients are symmetric around zero under these symmetry assumptions a locally asymptotically minimax adaptive estimator of the mean of the random coefficient is constructed the paper also proves the asymptotic normality of generalized m estimators of the parameter of interest these estimators are used as preliminary estimators in the above construction

grid based nonlinear filtering revisited recursive estimation asymptotic optimality this paper revisits grid based recursive approximate filtering of general markov processes in discrete time partially observed in conditionally gaussian noise the grid based filters considered rely on two types of state quantization namely the markovian type and the marginal type a set of novel relaxed sufficient conditions is proposed ensuring strong and fully characterized pathwise convergence of these filters to the respective mmse state estimator in particular for marginal state quantizations the notion of conditional regularity of stochastic kernels is introduced which to the best of the authors knowledge constitutes the most relaxed condition under which asymptotic optimality of the respective grid based filters is guaranteed further the convergence results are extended to include filtering of bounded and continuous functionals of the state as well as recursive approximate state prediction for both markovian and marginal quantizations the whole development of the respective grid based filters relies more on linear algebraic techniques and less on measure theoretic arguments making the presentation considerably shorter and technically simpler

estimation of noisy telegraph processes nonlinear filtering versus nonlinear smoothing corresp in the estimation problem of a two state stationary markov process with gaussian white noise added the optimal smoother is a two filter smoother in a special case the performance of the optimal nonlinear filter and smoother is evaluated analytically some asymptotic results are also derived

markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

state estimation in stochastic hybrid systems with sparse observations in this note we study the problem of state estimation for a class of sampled measurement stochastic hybrid systems where the continuous state x satisfies a linear stochastic differential equation and noisy measurements y are taken at assigned discrete time instants the parameters of both the state and measurement equation depend on the discrete state q of a continuous time finite markov chain even in the fault detection setting we consider at most one transition for q is admissible the switch may occur between two observations whence it turns out that the optimal estimates cannot be expressed in parametric form and time integrations are unavoidable so that the known estimation techniques cannot be applied we derive and implement an algorithm for the estimation of the states x q and of the discrete state switching time that is convenient for both recursive update and the eventual numerical quadrature numerical simulations are illustrated

performance bound and trellis code design criterion for discrete memoryless channels and finite delay symbol by symbol decoding a new tight analytical upper bound is presented for the performance of finite delay symbol by symbol sbs abend fritchman like decoding of trellis encoded data crossing arbitrary eventually time varying discrete memoryless channels dmcs on the basis of this bound a related criterion for the optimal design of trellis codes with sbs decoding is then proposed it gives rise to an effective procedure for the construction of good trellis codes generally time varying and nonlinear with assigned decoding complexity

jump diffusion processes in tracking recognition advances in sensor technology permit more sophisticated tracking identification algorithms to be implemented this correspondence compares and contrasts the modeling framework employed in two recent image enhanced trackers and generalizes one of them the pme for use in target identification

testing linear hypotheses in autoregressions the problem of testing linear hypotheses about the parameter vector of an autoregressive process with finite order is considered based on the property of local asymptotic normality we derive asymptotically optimal statistical tests additionally we define and investigate so called residual rank tests for these tests we obtain under the null hypothesis an asymptotic distribution which does not depend on the distribution of the innovation

filtering with uncertain noise filtering is concerned with estimating signals observed in noise in this technical note we consider a signal which is a markov chain in turn this is observed in gaussian noise whose parameters are not known peng has considered nonlinear expectations which can be represented as the supremum of a family of expectations this approach is followed below providing one of the first studies of filtering in this framework by extending methods of elliott et al modified recursive filters are derived where expected values are replaced by maximum likelihoods

state estimation for discrete time markov jump linear systems with time correlated measurement noise abstract in this paper the state estimation problem for discrete time markov jump linear systems affected by time correlated measurement noise is considered where the time correlated measurement noise is described by a linear system model with white noise as a result two algorithms are proposed to estimate the state of the system under consideration based on a measurement sequence the first algorithm is optimal in the sense of minimum mean square error which is obtained based on the measurement differencing method bayes rule and some results derived in this paper the second algorithm is a suboptimal algorithm obtained by using a lot of gaussian hypotheses the proposed suboptimal algorithm is finite dimensionally computable and does not increase computational and storage load with time computer simulations are carried out to evaluate the performance of the proposed suboptimal algorithm

general smoothing formulas for markov modulated poisson observations in this paper we compute general smoothing dynamics for partially observed dynamical systems generating poisson observations we consider two model classes each markov modulated poisson processes whose stochastic intensities depend upon the state of an unobserved markov process in one model class the hidden state process is a continuously valued ito spl circ process which gives rise to a continuous sample path stochastic intensity in the other model class the hidden state process is a continuous time markov chain giving rise to a pure jump stochastic intensity to compute filtered estimates of state process we establish dynamics whose solutions are unnormalized marginal probabilities however these dynamics include lebesgue stieltjes stochastic integrals by adapting the transformation techniques introduced by j m c clark we compute filter dynamics which do not include these stochastic integrals to construct smoothers we exploit a duality between our forward and backward transformed dynamics and thereby completely avoid the technical complexities of backward evolving stochastic integral equations the general smoother dynamics we present can readily be applied to specific smoothing algorithms referred to in the literature as fixed point smoothing fixed lag smoothing and fixed interval smoothing it is shown that there is a clear motivation to compute smoothers via transformation techniques similar to those presented by j m c clark that is our smoothers are easily obtained without recourse to two sided stochastic integration a computer simulation is included

on adaptive estimation in stationary arma processes on demontre la normalite asymptotique locale puis on construit des estimateurs localement asymptotiquement minimax

filters for estimating markov modulated poisson processes and image based tracking we present algorithms for state and parameter estimation of markov modulated poisson processes mmpp we first derive finite dimensional innovations and zakai filters for various statistics of a mmpp using these filters a filter based expectation maximization algorithm is derived for computing maximum likelihood parameter estimates finally we present an application of the techniques in image based tracking by estimating the state of a markovian jump linear system

reproducing gaussian densities and linear gaussian detection abstract it is shown that the expected value of a gaussian density function of a gaussian random variable is another gaussian density function this is used to determine which of m partially observed gaussian linear systems is the most likely given the observations

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

exact finite dimensional filters for maximum likelihood parameter estimation of continuous time linear gaussian systems in this paper we derive a new class of finite dimensional filters for integrals and stochastic integrals of moments of the state for continuous time linear gaussian systems apart from being of significant mathematical interest these new filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the model parameters

optimal terminal wealth under partial information both the drift and the volatility driven by a discrete time markov chain we consider a multistock market model the stock price process satisfies a stochastic differential equation where both the drift and the volatility are driven by a discrete time markov chain of finite states not only the underlying brownian motion but also the markov chain in the stochastic differential equation are assumed to be unobservable investors can observe the stock price process only the main result of this paper is that we derive the approximation of the optimal trading strategy and the corresponding optimal expected utility function from the terminal wealth for the crra utility function

pucci s conjecture and the alexandrov inequality for elliptic pdes in the plane the inequality of alexandrov bakel man and pucci is a basic tool in the theory of linear elliptic partial differential equations pdes which are not in divergence form as well as in the more general theory of nonlinear elliptic pdes here in two dimensions we prove the sharp form of the maximum principle as conjectured by pucci in 1966 give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting these results make use of the substantial recent advances in the planar theory of quasiconformal mappings astala was supported in part by the academy of finland projects 34082 and 41933 iwaniec by the national science foundation grants dms 0301582 and dms 0244297 martin by the n z marsden fund and the n z royal society james cook fellowship ams 1991 classification primary 30c60 35j15 35j70

asymptotically optimal discrete time nonlinear filters from stochastically convergent state process approximations we consider the problem of approximating optimal in the mmse sense nonlinear filters in a discrete time setting exploiting properties of stochastically convergent state process approximations more specifically we consider a class of nonlinear partially observable stochastic systems comprised by a possibly nonstationary hidden stochastic process the state observed through another conditionally gaussian stochastic process the observations under general assumptions we show that given an approximating process which for each time step is stochastically convergent to the state process in some appropriate sense an approximate filtering operator can be defined which converges to the true optimal nonlinear filter of the state in a strong and well defined sense i e compactly in time and uniformly in a completely characterized measurable set of probability measure almost unity also providing a purely quantitative justification of egoro s theorem for the problem at hand the results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating a large class of optimal nonlinear filters such as approximate grid based techniques known to perform well in a variety of applications

an m ary detection approach for asset allocation abstract we develop a continuous time asset allocation model which incorporates both model uncertainty and structural changes in economic conditions a dynamic m ary detection framework for a continuous time hidden markov chain partially observed in a gaussian process is used to model the price dynamics of the risky asset and the hidden states of an economy the goal of an investor is to select an optimal asset portfolio mix so as to maximize the expected utility of terminal wealth filtering theory is used first to turn the problem into one with complete observations and then to derive m ary detection filters for the hidden system the hamilton jacobi bellman dynamic programming approach is used to solve the asset allocation problem with complete observations an explicit solution is obtained for the power utility case

modal estimation in hybrid systems in hybrid systems estimation is made difficult by the nonlinear equations of state evolution several algorithms for generating an approximation to the conditional mean estimator have been proposed these are satisfactory in many cases but because of the way the observations are fused the estimate of the modal component of the state is often less good this paper proposes a novel algorithm for state estimation and compares it with some in current use

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

robust state estimation and fault diagnosis for uncertain hybrid systems abstract robust state estimation and fault diagnosis are challenging problems in the research into hybrid systems in this paper a novel robust hybrid observer is proposed for a class of hybrid systems with unknown inputs and faults model uncertainties disturbances and faults are represented as structured unknown inputs the robust hybrid observer consists of a mode observer for mode identification and a continuous observer for continuous state estimation and mode transition detection it is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded robustness to model uncertainties and disturbances can be guaranteed for the hybrid observer by disturbance decoupling furthermore the detectability and mode identifiability conditions are rigorously analyzed on the basis of the robust hybrid observer a robust fault detection and isolation scheme is presented also in the paper simulations of a hybrid four tank system show the proposed approach is effective

state and mode estimation for discrete time jump markov systems in this article we compute new state and mode estimation algorithms for discrete time gauss markov models whose parameter sets switch according to a known markov law an important feature of our algorithms is that they are based upon the exact filter dynamics computed in r j elliott f dufour and d sworder ieee trans automat control 41 1996 pp 1807 1810 the fundamental and well known obstacle in estimation of jump markov systems is managing the geometrically growing history of candidate hypotheses in our scheme we address this issue by proposing an extension of an idea due to viterbi our scheme maintains a fixed number of candidate paths in a history each identified by an optimal subset of estimated mode probabilities we compute finite dimensional suboptimal filters and smoothers which estimate the hidden state process and the mode probability our smoothers are based upon a duality between forward and backward dynamics further our smoothing algorithms are general and can be configured into the standard forms of fixed point fixed lag and fixed interval smoothers a computer simulation is included to demonstrate performance

maximum likelihood estimation of transition probabilities of jump markov linear systems this paper describes an online maximum likelihood estimator for the transition probabilities associated with a jump markov linear system jmls the maximum likelihood estimator is derived using the reference probability method which exploits an hypothetical probability measure to find recursions for complex expectations expectation maximization em procedure is utilized for maximizing the likelihood function in order to avoid the exponential increase in the number of statistics of the optimal em algorithm we make interacting multiple model imm type approximations the resulting method needs the mode weights of an imm filter with n3 components where n is the number of models in the jmls the algorithm can also supply base state estimates and covariances as a by product the performance of the estimator is illustrated on two simulated examples and compared to a recently proposed alternative

discrete time expectation maximization algorithms for markov modulated poisson processes in this paper we consider parameter estimation markov modulated poisson processes via robust filtering and smoothing techniques using the expectation maximization algorithm framework our filters and smoothers can be applied to estimate the parameters of our model in either an online configuration or an offline configuration further our estimator dynamics do not involve stochastic integrals and our new formulas in terms of time integrals are easily discretized and are written in numerically stable forms in w p malcolm r j elliott and j van der hoek ldquoon the numerical stability of time discretized state estimation via clark transformations rdquo presented at the ieee conf decision control mauii hi dec 2003

new finite dimensional filters and smoothers for noisily observed markov chains new finite dimensional filters and smoothers that are related to the wonham filter of a noisily observed markov chain are obtained in particular finite dimensional recursive filters and smoothers are given for the number of jumps from state i to state j for the occupation time of state i and for a stochastic integral related to the drift in the observations these filters allow easy application of the em algorithm for the estimation of the parameters of the markov chain and observation process

sequential monte carlo methods to train neural network models we discuss a novel strategy for training neural networks using sequential monte carlo algorithms and propose a new hybrid gradient descent sampling importance resampling algorithm hysir in terms of computational time and accuracy the hybrid sir is a clear improvement over conventional sequential monte carlo techniques the new algorithm may be viewed as a global optimization strategy that allows us to learn the probability distributions of the network weights and outputs in a sequential framework it is well suited to applications involving on line nonlinear and nongaussian signal processing we show how the new algorithm outperforms extended kalman filter training on several problems in particular we address the problem of pricing option contracts traded in financial markets in this context we are able to estimate the one step ahead probability density functions of the options prices

on the alexandroff bakelman pucci estimate and the reversed holder inequality for solutions of elliptic and parabolic equations the constant appearing in the classical alexandroff bakelman pucci estimate for subsolutions of second order uniformly elliptic equations in nondivergence form was known to depend on the diameter of the domain using the krylov safonov boundary weak harnack inequality due to trudinger we show that the dependence on the diameter may be replaced by dependence on a more precise geometric quantity of the domain as a consequence we get dependence on the measure instead of the diameter we also give new bounds for subsolutions in some unbounded domains such as domains contained in cones we apply the fabes and stroock reversed holder inequality for the green s function to improve our estimates we also give a new proof of the reversed holder inequality for the green s function based on the krylov safonov harnack inequality finally we find new bounds for subsolutions of uniformly parabolic equations in cylindrical and noncylindrical domains the constant in the alexandroff bakelman pucci krylov tso estimate was known to depend on the diameter of the base of the cylinder we get dependence either on the measure of the base or on the height of the cylinder we also give bounds for subsolutions in noncylindrical domains 1995 john wiley sons inc

state estimation for discrete time markov jump linear systems with time correlated and mode dependent measurement noise abstract the state estimation problem for discrete time markov jump linear systems corrupted by time correlated and mode dependent measurement noise is considered where the time correlated and mode dependent measurement noise is described via a discrete time stochastic system with markov parameter and kronecker delta function by defining the measurement noise in this manner both time correlation and periodic step change caused by the change of system environment or structure can be embodied in the measurement noise a novel distributed measurement differencing method is applied to the problem of state estimation under consideration so that two algorithms are obtained using some results presented in this paper the first algorithm is optimal in the sense of minimum mean square error which can exactly compute the minimum mean square error estimate of system state the second algorithm is suboptimal and the suboptimality of the algorithm is caused by using some gaussian hypotheses the two proposed algorithms are recursive and the proposed suboptimal algorithm has a time independent complexity the performance of the proposed suboptimal algorithm is illustrated using computer simulations

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

active classification for pomdps a kalman like state estimator author s zois ds levorato m mitra u abstract 1991 2012 ieee the problem of state tracking with active observation control is considered for a system modeled by a discrete time finite state markov chain observed through conditionally gaussian measurement vectors the measurement model statistics are shaped by the underlying state and an exogenous control input which influence the observations quality exploiting an innovations approach an approximate minimum mean squared error mmse filter is derived to estimate the markov chain system state to optimize the control strategy the associated mean squared error is used as an optimization criterion in a partially observable markov decision process formulation a stochastic dynamic programming algorithm is proposed to solve for the optimal solution to enhance the quality of system state estimates approximate mmse smoothing estimators are also derived finally the performance of the proposed framework is illustrated on the problem of physical activity detection in wireless body sensing networks the power of the proposed framework lies within its ability to accommodate a broad spectrum of active classification applications including sensor management for object classification and tracking estimation of sparse signals and radar scheduling

brief paper risk sensitive filtering for jump markov linear systems in this paper a risk sensitive multiple model filtering algorithm is derived using the reference probability methods first the approximation of the interacting multiple model imm algorithm is identified in the reference probability domain then the same type of approximation is used to derive the finite dimensional risk sensitive filtering algorithm the derived algorithm reduces to the imm filter when the risk sensitive parameter goes to zero and reduces to the risk sensitive filter for linear gauss markov systems when the number of models is unity the algorithm performs better in a simulated uncertain parameter scenario than the imm filter

optimal portfolios with maximum value at risk constraint under a hidden markovian regime switching model this paper studies an optimal portfolio selection problem in the presence of the maximum value at risk mvar constraint in a hidden markovian regime switching environment the price dynamics of n risky assets are governed by a hidden markovian regime switching model with a hidden markov chain whose states represent the states of an economy we formulate the problem as a constrained utility maximization problem over a finite time horizon and then reduce it to solving a hamilton jacobi bellman hjb equation using the separation principle the mvar constraint for n risky assets plus one riskless asset is derived and the method of lagrange multiplier is used to deal with the constraint a numerical algorithm is then adopted to solve the hjb equation numerical results are provided to demonstrate the implementation of the algorithm

a filtered em algorithm for joint hidden markov model and sinusoidal parameter estimation derives finite dimensional discrete time filters for estimating the parameters of discrete time finite state markov chains imbedded in a mixture of gaussian white noise and deterministic signals of known functional form with unknown parameters the filters that are derived estimate quantities used in the expectation maximization em algorithm for maximum likelihood ml estimation of the markov chain parameters transition probabilities and state levels as well as the parameters of the deterministic interference two types of deterministic signals are considered periodic or almost periodic signals with unknown frequency components amplitudes and phases and polynomial drift in the states of the markov process with the coefficients of the polynomial unknown the filter based em algorithm has negligible memory requirements in comparison implementing the em algorithm using smoothed variables forward backward variables requires memory proportional to the number of observations in addition the filters are suitable for multiprocessor implementation unlike the forward backward algorithm

on estimation and adaptive estimation for locally asymptotically normal families locally asymptotically minimax lam estimates are constructed for locally asymptotically normal lan families under very mild additional assumptions adaptive estimation is also considered and a sufficient condition is given for an estimate to be locally asymptotically minimax adaptive incidently it is shown that a well known lower bound due to hajek 1972 for the local asymptotic minimax risk is not sharp

efficient estimation of banach parameters in semiparametric models abtract consider a semiparametric model with a euclidean parameter and an infinite dimensional parameter to be called banach parameter assume there exists an efficient estimator of the euclidean parameter when the value of the euclidean parameter is known there exists an estimator of the banach parameter which depends on this value and is efficient within this restricted model substituting the efficient estimator of the euclidean parameter for the value of this parameter in the estimator of the banach parameter one obtains an efficient estimator of the banach parameter for the full semiparametric model with the euclidean parameter unknown this heredity property of efficiency completes estimation in semiparametric models in which the euclidean parameter has been estimated efficiently typically estimation of both the euclidean and the banach parameter is necessary in order to describe the random phenomenon under study to sufficient extent since efficient estimators are asymptotically linear the above substitution method is a particular case of substituting asymptotically linear estimators of a euclidean parameter into estimators that are asymptotically linear themselves and that depend on this euclidean parameter this more general substitution case is studied for its own sake as well and a heredity property for asymptotic linearity is proved

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

a high order markov switching model for risk measurement in this paper we introduce a high order markov switching hms model for measuring the risk of a portfolio we suppose that the rate of return from a risky portfolio follows an hms model with the drift and the volatility modulated by a discrete time weak markov chain the states of the weak markov chain are interpreted as observable states of an economy we adopt the value at risk var as a metric for market risk quantification and examine the high order effect of the underlying markov chain on the risk measures via backtesting

time discretization of continuous time filters and smoothers for hmm parameter estimation in this paper we propose algorithms for parameter estimation of fast sampled homogeneous markov chains observed in white gaussian noise our algorithms are obtained by the robust discretization of stochastic differential equations involved in the estimation of continuous time hidden markov models hmm s via the em algorithm we present two algorithms the first is based on the robust discretization of continuous time filters that were recently obtained by elliott to estimate quantities used in the em algorithm the second is based on the discretization of continuous time smoothers yielding essentially the well known baum welch re estimation equations the smoothing formulas for continuous time hmm s are new and their derivation involves two sided stochastic integrals the choice of discretization results in equations which are identical to those obtained by deriving the results directly in discrete time the filter based em algorithm has negligible memory requirements indeed independent of the number of observations in comparison the smoother based discrete time em algorithm requires the use of the forward backward algorithm which is a fixed interval smoothing algorithm and has memory requirements proportional to the number of observations on the other hand the computational complexity of the filter based em algorithm is greater than that of the smoother based scheme however the filters may be suitable for parallel implementation using computer simulations we compare the smoother based and filter based em algorithms for hmm estimation we provide also estimates for the discretization error

comment on adaptive estimation in time series regression models by d g steigerwald abstract in this comment we discuss some serious shortcomings and errors in steigerwald 1992

adaptive estimation in time series regression models this work develops adaptive estimators for a linear regression model with serially correlated errors we show that these results continue to hold when the order of the arma process characterizing the errors is unknown the finite sample results are promising indicating that substantial efficiency gains may be possible for samples as small as 50 observations we use these estimators to investigate the behavior of the forward foreign exchange market

exact filters for the estimation of the number of transitions of finite state continuous time markov processes the problem of estimating the number of transitions of finite state continuous time markov processes observed by a noisy sensor is considered a finite dimensional exact filter is derived and using the em algorithm an extension of the baum welch algorithm for the discrete time case an application is made to the problem of estimating the unknown transition matrix of a finite state continuous time markov process

exact hybrid filters in discrete time using the reference probability method and the change of measure in discrete time the state estimator problem is considered for linear systems observed in gaussian noise when the coefficients are functions of a noisily observed finite state markov chain the methods are new and finite dimensional filters are obtained however the number of statistics increases in time a numerical comparison of this filter with the interactive multiple model algorithm introduced by blom and bar shalom 1988 is given

markowitz s mean variance portfolio selection with regime switching from discrete time models to their continuous time limits we study a discrete time version of markowitz s mean variance portfolio selection problem where the market parameters depend on the market mode regime that jumps among a finite number of states the random regime switching is delineated by a finite state markov chain based on which a discrete time markov modulated portfolio selection model is presented such models either arise from multiperiod portfolio selections or result from numerical solution of continuous time problems the natural connections between discrete time models and their continuous time counterpart are revealed since the markov chain frequently has a large state space to reduce the complexity an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a two time scale problem we prove that the process of interest yields a switching diffusion limit using weak convergence methods next based on the optimal control of the limit process obtained from our recent work we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality

a filter for a hidden markov chain observed in fractional gaussian noise abstract estimates of parameters and hidden states in a hidden markov model where the noise in the observations is fractional gaussian noise are obtained

estimating invariant laws of linear processes by u statistics suppose we observe an invertible linear process with independent mean zero innovations and with coefficients depending on a finite dimensional parameter and we want to estimate the expectation of some function under the stationary distribution of the process the usual estimator would be the empirical estimator it can be improved using the fact that the innovations are centered we construct an even better estimator using the representation of the observations as infinite order moving averages of the innovations then the expectation of the function under the stationary distribution can be written as the expectation under the distribution of an infinite series in terms of the innovations and it can be estimated by a u statistic of increasing order also called an infinite order u statistic in terms of the estimated innovations the estimator can be further improved using the fact that the innovations are centered this improved estimator is optimal if the coefficients of the linear process are estimated optimally the variance reduction of our estimator over the empirical estimator can be considerable

conditional densities for continuous time nonlinear hybrid systems with applications to fault detection continuous time nonlinear stochastic differential state and measurement equations all of which have coefficients capable of abrupt changes at a random time are considered finite state jump markov chains are used to model the changes conditional probability densities which are essential in obtaining filtered estimates for these hybrid systems are then derived they are governed by a coupled system of stochastic partial differential equations when the q matrix of the markov chain is either lower or upper diagonal it is shown that the system of conditional density equations is finite dimensional computable these findings are then applied to a fault detection problem to compute state estimates that include the failure time

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

on hybrid state estimation for stochastic hybrid systems this paper considers the state estimation problem for the general continuous time stochastic hybrid system shs which has various applications defined on the hybrid state space the shs has the interacting discrete dynamics and continuous dynamics subject to various uncertainties the hybrid state estimation problem is to estimate both the continuous state and the discrete state of the shs with the information given by a continuous time observation process in this paper the hybrid state estimation problem is mathematically formulated and the corresponding filtering equations that are stochastic partial differential equations are derived to describe the evolution of the hybrid state estimates conditioned on the observation history a numerical algorithm based on a finite difference approach is proposed to solve the filtering equations a markov chain mc is constructed on the descretized hybrid state space to approximate the infinitesimal generator of the shs and then hybrid state estimation for the shs is reduced to estimating the state of the mc it is proved that the state estimation results of the mc converge to the solution to the filtering equations as the constructed mc converges to the shs an illustrative example of aircraft tracking is used to demonstrate the performance of the proposed algorithm

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

discrete time estimation of a markov chain with marked point process observations application to markovian jump filtering in this paper various discrete time estimation problems are studied for a finite and homogeneous markov chain observed by a marked point process these problems which could have significant applications in target tracking manufacturing or communication theory have never been studied in the literature the quantities to be estimated are the state the number of jumps and the occupation times the identification of the chain transition matrix is also addressed via an expectation maximization procedure solutions in the sense of the conditional distribution are obtained by a change of probability measure and are shown to have convenient recursive forms the efficiency of this new approach for sensor modeling is illustrated by the study of a linear markovian jump filtering problem where in addition to a classical state observation a mode markov point process observation is assumed a numerical example is given

new finite dimensional filters for parameter estimation of discrete time linear gaussian models the authors derive a new class of finite dimensional recursive filters for linear dynamical systems the kalman filter is a special case of their general filter apart from being of mathematical interest these new finite dimensional filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the parameters of a linear dynamical system important advantages of their filter based em algorithm compared with the standard smoother based em algorithm include 1 substantially reduced memory requirements and 2 ease of parallel implementation on a multiprocessor system the algorithm has applications in multisensor signal enhancement of speech signals and also econometric modeling

asymptotic minimax estimation in semiparametric models we give several conditions on the estimator of efficient score function for estimating the parametric component of semiparametric models a semiparametric version of the one step mle using an estimator of efficient score function which fulfills the conditions is shown to converge to the normal distribution with minimum variance locally uniformly over a fairly large neighborhood around the assumed semiparametric model consequently it is shown to be asymptotically minimax with bounded subconvex loss functions a few examples are also considered

semiparametrically efficient inference based on signed ranks in symmetric independent component models we consider semiparametric location scatter models for which the p variate observation is obtained as x lambda z mu where mu is a p vector lambda is a full rank p p matrix and the unobserved random p vector z has marginals that are centered and mutually independent but are otherwise unspecified as in blind source separation and independent component analysis ica the parameter of interest throughout the paper is lambda on the basis of n i i d copies of x we develop under a symmetry assumption on z signed rank one sample testing and estimation procedures for lambda we exploit the uniform local and asymptotic normality ulan of the model to define signed rank procedures that are semiparametrically efficient under correctly specified densities yet as usual in rank based inference the proposed procedures remain valid correct asymptotic size under the null for hypothesis testing and root n consistency for point estimation under a very broad range of densities we derive the asymptotic properties of the proposed procedures and investigate their finite sample behavior through simulations

efficiency improvements in inference on stationary and nonstationary fractional time series we consider a time series model involving a fractional stochastic component whose integration order can lie in the stationary invertible or nonstationary regions and be unknown and an additive deterministic component consisting of a generalized polynomial the model can thus incorporate competing descriptions of trending behavior the stationary input to the stochastic component has parametric autocorrelation but innovation with distribution of unknown form the model is thus semiparametric and we develop estimates of the parametric component which are asymptotically normal and achieve an m estimation efficiency bound equal to that found in work using an adaptive lam lan approach a major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies this is relevant also when the innovation density is parameterized and we provide a result for that case also our semiparametric estimates employ nonparametric series estimation which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models our work thus also contributes to methods and theory for nonfractional time series models such as autoregressive moving averages a monte carlo study of finite sample performance of the semiparametric estimates is included

image enhanced multiple model tracking we consider tracking algorithms for manoeuvring targets when the observations include extra information on the current operating mode of the target obtained from an image sensor the target is modelled as a markov jump linear system and the image based observations form a discrete time point process we derive the optimal minimum mean square error filtered estimate which intrinsically fuses the image based and primary observations this optimal filter is computationally prohibitive but provides the basis for a clear understanding of various suboptimal approaches we propose the image enhanced imm filter as a practical alternative which retains many desirable properties of the optimal filter and outperforms existing image enhanced tracking algorithms over a broad range of operating scenarios

kalman filtering for linear systems with coefficients driven by a hidden markov jump process abstract we present an explicit algorithm for the solution of the nonlinear filtering problem concerning a linear partially observed diffusion type model where the coefficients are driven by a hidden markov jump process of which one can observe the occurrence of a jump the recursive filter is of a branching type with its finite dimension growing for each jump of the hidden markov process

bond pricing under a markovian regime switching jump augmented vasicek model via stochastic flows in this article we shall explore the state of art of stochastic flows to derive an exponential affine form of the bond price when the short rate process is governed by a markovian regime switching jump diffusion version of the vasicek model we provide the flexibility that the market parameters including the mean reversion level the volatility rate and the intensity of the jump component switch over time according to a continuous time finite state markov chain the states of the chain may be interpreted as different states of an economy or different stages of a business cycle we shall provide a representation for the exponential affine form of the bond price in terms of fundamental matrix solutions of linear matrix differential equations

a finite dimensional filter for hybrid observations the paper considers a system whose dynamics are described in terms of the hybrid of a var 1 type process and discrete state space markov chain the paper provides a recursive expression for the conditional density distributions of these two processes given a set of observations which are a noisy linear combination of these two processes

multiple model methods in path following ii path following is difficult when the observation rate is low multiple model estimators incorporating multisensor fusion have proven useful in this application this paper shows the advantage of a recently developed multiple model algorithm performance comparisons with some current algorithms are presented

robust continuous time smoothers without two sided stochastic integrals we consider the problem of fixed interval smoothing of a continuous time partially observed nonlinear stochastic dynamical system existing results for such smoothers require the use of two sided stochastic calculus the main contribution of the paper is to present a robust formulation of the smoothing equations under this robust formulation the smoothing equations are nonstochastic parabolic partial differential equations with random coefficients and hence the technical machinery associated with two sided stochastic calculus is not required furthermore the robust smoothed state estimates are locally lipschitz in the observations which is useful for numerical simulation as examples finite dimensional robust versions of the benes and hidden markov model smoothers and smoothers for piecewise linear dynamics are derived these finite dimensional smoothers do not involve stochastic integrals

exact adaptive filters for markov chains observed in gaussian noise abstract a discrete time finite state markov chain is observed through a real or vector valued function whose values are corrupted by gaussian noise by introducing a new measure exact unnormalized recursive estimates and smoothers are obtained for the state of the markov chain for the number of jumps from one state to another for the occupation time in any state and for processes related to the observation parameters using the em algorithm these allow estimates of all the parameters of the model to be revised including the variance of the gaussian noise in the observations the filters are therefore adaptive or self tuning

exact bayesian filter and joint imm coupled pda tracking of maneuvering targets from possibly missing and false measurements this paper represents the problem of tracking multiple maneuvering targets from possibly missing and false measurements as one of filtering for a jump linear descriptor system with stochastic i i d coefficients this particular representation serves as an instrument in the characterization of the exact bayesian filter subsequently novel finite dimensional filter algorithms are developed through introducing approximations to the exact bayesian solution one filter approximation assumes conditionally gaussian density of the joint target state given the joint target maneuver mode and the algorithm is referred to as joint imm coupled pda jimmcpda the specialty of this filter algorithm is that both the imm step and the pda step are performed jointly over the modes and states of all targets subsequently the cpda track coalescence avoiding hypothesis pruning approach of blom bloem 2000 probabilistic data association avoiding track coalescence ieee transactions of automatic control 45 247 259 is extended to bring the joint target modes into account the resulting filter algorithm is referred to as track coalescence avoiding joint imm coupled pda the two novel algorithms are compared to immjpda and immpda through monte carlo simulations

robust m ary detection filters and smoothers for continuous time jump markov systems in this paper we consider a dynamic m ary detection problem when markov chains are observed through a wiener process these systems are fully specified by a candidate set of parameters whose elements are a rate matrix for the markov chain and a parameter for the observation model further we suppose these parameter sets can switch according to the state of an unobserved markov chain and thereby produce an observation process generated by time varying jump stochastic parameter sets given such an observation process and a specified collection of models we estimate the probabilities of each model parameter set explaining the observation by defining a new augmented state process then applying the method of reference probability we compute matrix valued dynamics whose solutions estimate joint probabilities for all combinations of candidate model parameter sets and values taken by the indirectly observed state process these matrix valued dynamics satisfy a stochastic integral equation with a wiener process integrator using the gauge transformation techniques introduced by clark and a pointwise matrix product we compute robust matrix valued dynamics for the joint probabilities on the augmented state space in these new dynamics the observation wiener process appears as a parameter matrix in a linear ordinary differential equation rather than an integrator in a stochastic integral equation it is shown that these robust dynamics when discretised enjoy a deterministic upper bound which ensures nonnegative probabilities for any observation sample path in contrast no such upper bounds can be computed for taylor expansion approximations such as the euler maryauana and milstein schemes finally by exploiting a duality between causal and anticausal robust detector dynamics we develop an algorithm to compute smoothed mode probability estimates without stochastic integrations a computer simulation demonstrating performance is included

a stochastic maximum principle for a markov regime switching jump diffusion model and its application to finance this paper develops a sufficient stochastic maximum principle for a stochastic optimal control problem where the state process is governed by a continuous time markov regime switching jump diffusion model we also establish the relationship between the stochastic maximum principle and the dynamic programming principle in a markovian case applications of the stochastic maximum principle to the mean variance portfolio selection problem are discussed

adaptive estimation in time series models in a framework particularly suited for many time series models we obtain a lan result under quite natural and economical conditions this enables us to construct adaptive estimators for part of the euclidean parameter in these semiparametric models special attention is directed to group models in time series with the important subclass of models with time varying location and scale our set up is confronted with the existing literature and as examples we reconsider linear regression and arma tar and arch models