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

empirical processes typical sequences and coordinated actions in standard borel spaces this paper proposes a new notion of typical sequences on a wide class of abstract alphabets so called standard borel spaces which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable test functions in the finite alphabet case we can take all uniformly bounded functions and recover the usual notion of strong typicality or typicality under the total variation distance for a general alphabet however this function class turns out to be too large and must be restricted with this in mind we define typicality with respect to any glivenko cantelli function class i e a function class that admits a uniform law of large numbers and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios in which the relevant operational criteria pertain to reproducing empirical averages of a general alphabet stationary memoryless source with respect to a suitable function class

contractive maps on operator ideals and norm inequalities ii abstract let i be a norm ideal of operators equipped with a unitarily invariant norm we exploit integral representations of certain functions to prove that certain ratios of linear operators acting on operators in i are contractive this leads to some new and old norm inequalities we also lift a variety of inequalities to the operator setting which were proved in the matrix setting earlier

on nearly optimal controls of hybrid lqg problems the paper is concerned with hybrid control of a class of linear quadratic gaussian lqg systems modulated by a finite state markov chain it develops approximation schemes for systems involving singularly perturbed markov chains with weak and strong interactions which are useful for natural time scale separation and for large scale markovian systems the presentation encompasses three cases the recurrent markov chains inclusion of transient states and inclusion of absorbing states near optimality of the approximation schemes is obtained and a numerical example is presented computational results indicate our approximation schemes perform well

robust jump linear quadratic control a mode stabilizing solution this correspondence is concerned with optimal feedback control of a class of linear stochastic systems for a quadratic performance index the solution to this problem is known but the resulting optimal regulator does not ensure stability in every mode of operation to improve robustness a new mode stabilizing solution is defined an example illustrates the advantages of the proposed controller

the empirical distribution of rate constrained source codes let x x sub 1 be a stationary ergodic finite alphabet source x sup n denote its first n symbols and y sup n be the codeword assigned to x sup n by a lossy source code the empirical kth order joint distribution q spl circ sup k x sup n y sup n spl rceil x sup k y sup k is defined as the frequency of appearances of pairs of k strings x sup k y sup k along the pair x sup n y sup n our main interest is in the sample behavior of this random distribution letting i q sup k denote the mutual information i x sup k y sup k when x sup k y sup k spl sim q sup k we show that for any sequence of lossy source code s of rate spl les r lim sup sub n spl rarr spl infin 1 k i q spl circ sup k x sup n y sup n spl rfloor spl les r 1 k h x sub 1 sup k h x a s where h x denotes the entropy rate of x this is shown to imply for a large class of sources including all independent and identically distributed i i d sources and all sources satisfying the shannon lower bound with equality that for any sequence of codes which is good in the sense of asymptotically attaining a point on the rate distortion curve q spl circ sup k x sup n y sup n spl rfloor spl rarr sup d p x sup k y sup k a s whenever p sub x sup k sub y sup k is the unique distribution attaining the minimum in the definition of the kth order rate distortion function consequences of these results include a new proof of kieffer s sample converse to lossy source coding as well as performance bounds for compression based denoisers

matrix young inequalities for the hilbert schmidt norm abstract let a b and x be n n complex matrices such that a and b are positive semidefinite if p q 1 with 1 p 1 q 1 it is shown that 1 p a p x 1 q xb q 2 2 1 r 2 a p x xb q 2 2 axb 2 2 where r max p q and 2 is the hilbert schmidt norm generalizations and applications of this inequality are also considered

a fault detection observer design for lpv systems in finite frequency domain this paper addresses the fault detection observer design problem for linear parameter varying systems two finite frequency performance indexes are introduced to measure the fault sensitivity and the disturbance robustness first the h index fault sensitivity condition in finite frequency domain is obtained by generalised kalman yakubovich popov lemma and new linearisation techniques then with the aid of kalman yakubovich popov lemma and projection lemma the stability and robustness conditions are derived it turns out that the non convexity problem which is caused by dealing with the above three conditions can be translated into a bilinear matrix inequality optimisation problem by increasing the dimensions of slack variable matrix an iterative linear matrix inequality algorithm is proposed to get the solution the effectiveness of the filter is shown via three numerical examples

norm inequalities in operator ideals abstract in this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm among the well known inequalities which can be proved with this technique are the lowner heinz inequality inequalities relating various operator means and the corach porta recht inequality we prove two general inequalities and from them we derive several inequalities by specialization many of them new we also show how some inequalities known to be valid for matrices or bounded operators can be extended with this technique to normed ideals in c algebras in particular to the noncommutative l p spaces of a semi finite von neumann algebra

more accurate operator means inequalities abstract our main target in this paper is to present new sharp bounds for inequalities that result when weighted operator means are filtered through positive linear maps and operator monotone functions as an application we prove a refined reverse of the celebrated golden thompson inequality furthermore we show how these inequalities can be squared

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

arithmetic geometric mean and related inequalities for operators abstract in recent years certain arithmetic geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on we first point out that they are direct consequences of integral expressions of relevant operators furthermore we obtain related new inequalities theorems 4 5 and 6 based on our current approach

a cauchy schwarz inequality for operators with applications abstract for any unitarily invariant norm on hilbert space operators it is shown that for all operators a b x and positive real numbers r we have a xb r 2 aa x r xbb r some consequences are then discussed a simple proof is given for the fact that for positive operators a b the function spr a t b t 1 t is monotone in t on the positive half line

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

families of completely positive maps associated with monotone metrics abstract an operator convex function on 0 which satisfies the symmetry condition k x 1 x k x can be used to define a type of non commutative multiplication by a positive definite matrix or its inverse using the primitive concepts of left and right multiplication and the functional calculus the operators for the inverse can be used to define quadratic forms associated with riemannian metrics which contract under the action of completely positive trace preserving maps we study the question of when these operators define maps which are also completely positive cp although a d 1 2 a d 1 2 is the only case for which both the map and its inverse are cp there are several well known one parameter families for which either the map or its inverse is cp we present a complete analysis of the behavior of these families as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points our primary tool is an order relation based on the concept of positive definite functions although some results can be obtained from known properties we also prove new results based on the positivity of the fourier transforms of certain functions concrete computations of certain fourier transforms not only yield new examples of positive definite functions but also examples in the much stronger class of infinitely divisible functions

improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed

linear congruential trellis source codes design and analysis rate distortion trellis source codes are developed for quantizing memoryless independent and identically distributed i i d sources the codes are generated by simple linear congruential recursions the method generates codes at a variety of rates including fractional ones reproducer sets can be large a crucial advantage with certain sources axioms for good code construction are developed that are based on recursion properties and certain trellis symmetries these axioms are justified by the outcome of random searches for good codes it is found that the trellis code design breaks into two problems the trellis labels should have certain properties regardless of the source distribution and the reproducer values depend on the source encoders are simulated for a number of continuous amplitude sources for the same computational complexity the new codes in most cases perform better than the best codes in the literature including trellis coded quantization and fake process approaches

an arithmetic geometric harmonic mean inequality involving hadamard products abstract given matrices of the same size a a ij and b b ij we define their hadamard product to be a b a ij b ij we show that if x i 0 and q p 0 then the n n matrices x i x j x i x j x i 1 x j 1 x i x j and x i p x j p x i q x j q are positive definite and we relate these facts to some matrix valued arithmetic geometric harmonic mean inequalities some of which involve hadamard products and others unitarily invariant norms it is known that if a is positive semidefinite then max a b b 1 max a ij where denotes the spectral norm we show that the converse of this statement is false and give a useful partial converse

on infinite divisibility of positive definite functions arising from operator means abstract for a point wisely non negative positive definite function a certain criterion for its infinite divisibility i e all its fractional powers are also positive definite is obtained this criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means in particular we determine when the function cosh x s cosh x s 0 1 s s 1 1 is infinitely divisible

a finite frequency domain approach to fault detection for linear discrete time systems this paper deals with the fault detection fd problem in the finite frequency domain for linear time invariant discrete time systems with bounded disturbances a fault detection observer is designed by employing two performance indesxes which are used to increase the fault sensitivity in finite frequency domain and attenuate the effects of disturbances in full frequency domain with the aid of the generalized kalman yakubovich popov gkyp lemma the design methods are presented in terms of solutions to a set of linear matrix inequalities lmis numerical examples are given to illustrate the effectiveness of the proposed method

simulation of random processes and rate distortion theory we study the randomness necessary for the simulation of a random process with given distributions on terms of the finite precision resolvability of the process finite precision resolvability is defined as the minimal random bit rate required by the simulator as a function of the accuracy with which the distributions are replicated the accuracy is quantified by means of various measures variational distance divergence orstein 1973 prohorov 1956 and related measures of distance between the distributions of random process in the case of ornstein prohorov and other distances of the kantorovich vasershtein type we show that the finite precision resolvability is equal to the rate distortion function with a fidelity criterion derived from the accuracy measure this connection leads to new results on nonstationary rate distortion theory in the case of variational distance the resolvability of stationary ergodic processes is shown to equal entropy rate regardless of the allowed accuracy in the case of normalized divergence explicit expressions for finite precision resolvability are obtained in many cases of interest and connections with data compression with minimum probability of block error are shown

positive definite solutions of some matrix equations in this paper we investigate some existence questions of positive semi definite solutions for certain classes of matrix equations known as the generalized lyapunov equations we present sufficient and necessary conditions for certain equations and only sufficient for others

construction and evaluation of trellis coded quantizers for memoryless sources new constructions of trellis waveform coders trellis coded quantizers and trellis coded vector quantizers are proposed the performances of the new quantizers are determined for the memoryless laplacian gaussian and uniform sources they are better than for the gaussian and laplacian sources or equal to for the uniform source the best previously published results

a mapping approach to rate distortion computation and analysis in rate distortion theory results are often derived and stated in terms of the optimizing density over the reproduction space in this paper the problem is reformulated in terms of the optimal mapping from the unit interval with lebesgue measure that would induce the desired reproduction probability density this results in optimality conditions that are random relatives of the known lloyd 1982 optimality conditions for deterministic quantizers the validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces we show that for the squared error distortion the optimal reproduction random variable is purely discrete at supercritical distortion where the shannon 1948 lower bound is not tight the gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate to analyze the evolution of the optimal reproduction distribution we use the mapping formulation and establish an analogy to statistical mechanics the solutions are given by the distribution at isothermal statistical equilibrium and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate distortion theory the analysis of an annealing process shows how the number of symbols grows as the system undergoes phase transitions thus an algorithm based on the mapping approach often needs but a few variables to find the exact solution while the blahut 1972 algorithm would only approach it at the limit of infinite resolution finally a quick deterministic annealing algorithm to generate the rate distortion curve is suggested the resulting curve is exact as long as continuous phase transitions in the process are accurately followed

strong monotonicity for various means abstract point wise monotonicity in parameters for various one parameter families of scalar means such as power difference means binomial means and stolarsky means is well known but norm comparison for corresponding operator means requires monotonicity in the sense of positive definiteness among other things we obtain monotonicity in the sense of infinite divisibility which is much stronger than that in the sense of positive definiteness these strong monotonicity results are proved based on explicit computations for measures in relevant levy khintchine or actually kolmogorov formulas

near optimal controls of random switching lq problems with indefinite control weight costs in this paper we consider hybrid controls for a class of linear quadratic problems with white noise perturbation and markov regime switching where the regime switching is modeled by a continuous time markov chain with a large state space and the control weights are indefinite the use of large state space enables us to take various factors of uncertain environment into consideration yet it creates computational overhead and adds difficulties aiming at reduction of complexity we demonstrate how to construct near optimal controls first in the model we introduce a small parameter to highlight the contrast of the weak and strong interactions and fast and slow motions this results in a two time scale formulation in view of the recent developments on lq problems with indefinite control weights and two time scale markov chains we then establish the convergence of the system of riccati equations associated with the hybrid lq problem based on the optimal feedback control of the limit system obtained using the system of riccati equations we construct controls for the original problem and show that such controls are near optimal a numerical demonstration of a simple system is presented here

identification of wiener hammerstein and narx systems as markov chains with improved estimates for their nonlinearities abstract the wiener hammerstein and nonlinear arx systems are identified not only for their linear subsystems if they exist but also for the nonlinearities with their first derivatives it is assumed that the input signals and noises are mutually independent and both are sequences of independent and identically distributed iid random variables the estimates based on the stochastic approximation algorithms with expanding truncations saawet are proved to be strongly consistent with the help of the markov properties possessed by these systems the estimates of the first derivatives improve the accuracy of interpolating the nonlinearity curves as validated by simulation examples

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

an arithmetic geometric mean inequality for products of three matrices abstract consider the following noncommutative arithmetic geometric mean inequality given positive semidefinite matrices a 1 a n the following holds for each integer m n 1 n m j 1 j 2 j m 1 n a j 1 a j 2 a j m n m n j 1 j 2 j m 1 all distinct n a j 1 a j 2 a j m where denotes a unitarily invariant norm including the operator norm and schatten p norms as special cases while this inequality in full generality remains a conjecture we prove that the inequality holds for products of up to three matrices m 3 the proofs for m 1 2 are straightforward to derive the proof for m 3 we appeal to a variant of the classic araki lieb thirring inequality for permutations of matrix products

optimal mean variance control for discrete time linear systems with markovian jumps and multiplicative noises in this paper we consider the stochastic optimal control problem of discrete time linear systems subject to markov jumps and multiplicative noises under two criteria the first one is an unconstrained mean variance trade off performance criterion along the time and the second one is a minimum variance criterion along the time with constraints on the expected output we present explicit conditions for the existence of an optimal control strategy for the problems generalizing previous results in the literature we conclude the paper by presenting a numerical example of a multi period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

jump linear quadratic gaussian control in continuous time the optimal quadratic control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes is addressed the systems are also subject to gaussian input and measurement noise the optimal solution for the jump linear quadratic gaussian jlqc problem is given this solution is based on a separation theorem the optimal state estimator is sample path dependent if the plant parameters are constant in each value of the underlying jumping process then the controller portion of the compensator converges to a time invariant control law however the filter portion of the optimal infinite time horizon jlqc compensator is not time invariant thus a suboptimal filter which does converge to a steady state solution under certain conditions is derived and a time invariant compensator is obtained

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

lossy compression of discrete sources via the viterbi algorithm we present a new lossy compressor for finite alphabet sources for coding a sequence xn the encoder starts by assigning a certain cost to each possible reconstruction sequence it then finds the one that minimizes this cost and describes it losslessly to the decoder via a universal lossless compressor the cost of each sequence is a linear combination of its distance from the sequence xn and a linear function of its kth order empirical distribution the structure of the cost function allows the encoder to employ the viterbi algorithm to find the sequence with minimum cost we identify a choice of the coefficients used in the cost function which ensures that the algorithm universally achieves the optimum rate distortion performance for any stationary ergodic source in the limit of large provided that increases as o log n iterative techniques for approximating the coefficients which alleviate the computational burden of finding the optimal coefficients are proposed and studied

comparison of various means for operators for hilbert space operatorsh k xwithh k 0 the norm inequality h1 2xk1 2 12 hx xk is known where is an arbitrary unitarily invariant norm a refinement of this arithmetic geometric mean inequality is studied similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean

algorithms for singularly perturbed limiting average markov control problems a singularly perturbed markov decision process with the limiting average reward criterion is considered it is assumed that the underlying process is composed of n separate irreducible processes and that the small perturbation is such that it unites these processes into a single irreducible process two algorithms for the solution of the underlying limit markov control problem are presented the first of these is a linear program possessing the wolfe dantzig structure inherited from the ergodic nearly decomposable assumption in the model the second is an aggregation disaggregation policy improvement algorithm

on some operator inequalities abstract for hilbert space operators s t with s invertible and self adjoint corach porta and recht recently proved that sts 1 s 1 ts 2 t a generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic geometric mean inequality some related inequalities are also discussed

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

stochastic linear quadratic regulators with indefinite control weight costs this paper considers optimal minimizing control of stochastic linear quadratic regulators lqrs the assumption that the control weight costs must be positive definite inherited from the determ

cauchy schwarz norm inequalities for weak integrals of operator valued functions for a finite measures on and weakly measurable families at t and bt t of hilbert space operators we have the non commutative cauchy schwarz inequalities in schatten p ideals axbd p a aa d q 1ad 2qx b b bd r 1b d 2rp for all x p h and for all p q r 1 such that 1q 1r 2p if both at t and bt t consists of commuting normal operators then axbd a ad x b bd for all unitarily invariant norms and all x h if additionally a ad i and b bd i then i a ad xi b bd h and i a ad xi b bd x axbd applications include young s and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions

rate constrained simulation and source coding i i d sources necessary conditions for asymptotically optimal sliding block or stationary codes for source coding and rate constrained simulation of memoryless sources are presented and used to motivate a design technique for trellis encoded source coding and rate constrained simulation the code structure has intuitive similarities to classic random coding arguments as well as to fake process methods and alphabet constrained methods experimental evidence shows that the approach provides comparable or superior performance in comparison with previously published methods on common examples sometimes by significant margins

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

sliding block and random source coding with constrained size reproduction alphabets recent research has shown that nearly optimum ratedistortion function performance can be obtained in encoding memoryless continuous amplitude sources using a constrained size reproduction alphabet in this paper we use a four letter reproduction alphabet in encoding the gaussian and laplacian sources with the squared error distortion measure we introduce a sliding block code generator decoder and perform extensive coding simulations at a rate of one bit source symbol the average distortions in those simulations are much below the optimum quantization distortion and with much smaller search intensity are about the same or lower than distortions reported previously in comparable research the slidingblock code results are also compared to those of parallel random coding simulations with the same reproduction alphabet and rate

indefinite quadratic with linear costs optimal control of markov jump with multiplicative noise systems in this paper we consider the stochastic optimal control problem of discrete time markov jump with multiplicative noise linear systems the performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables the weighting matrices of the state and control for the quadratic part are allowed to be indefinite we present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized riccati difference equations interconnected with a set of coupled linear recursive equations for the case in which the quadratic term matrices are non negative this necessary and sufficient condition can be written in a more explicit way the results are applied to a problem of portfolio optimization

weak convergence of hybrid filtering problems involving nearly completely decomposable hidden markov chains concentrating on a class of hybrid discrete time filtering problems that are modulated by a markov chain this work aims to reduce the complexity of the underlying problems since the markov chain has a large state space the solution of the problem relies on solving a large number of filtering equations exploiting the hierarchical structure of the system it is noted that the transition probability matrix of the markov chain can be viewed as a nearly decomposable one it is shown that a reduced system of filtering equations can be obtained by aggregating the states of each recurrent class into one state extensions to inclusion of transient states and nonstationary cases are also treated