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

linear quadratic control of backward stochastic differential equations this paper is concerned with optimal control of linear backward stochastic differential equations bsdes with a quadratic cost criteria or backward linear quadratic blq control the solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion of squares technique two alternative though equivalent expressions for the optimal control are obtained the first of these involves a pair of riccati type equations an uncontrolled bsde and an uncontrolled forward stochastic differential equation sde while the second is in terms of a hamiltonian system contrary to the deterministic or stochastic forward case the optimal control is no longer a feedback of the current state rather it is a feedback of the entire history of the state a key step in our derivation is a proof of global solvability of the aforementioned riccati equations although of independent interest this issue has particular relevance to the blq problem since these riccati equations play a central role in our solution last but not least it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear quadratic lq problem this in turn reveals the origin of the riccati equations introduced

the maximum principle for partially observed optimal control problems of mean field fbsdes abstractthis paper is concerned with a partially observed optimal control problem described by mean field forward and backward stochastic differential equations moreover the control variable ente

accelerated solution of multivariate polynomial systems of equations we propose new las vegas randomized algorithms for the solution of a square nondegenerate system of equations with well separated roots the algorithms use oc delta csttn d 2 log d log b arithmetic operations in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal to approximate all real roots of the system as well as all roots lying in a fixed n dimensional box or disc here d is an upper bound on the number of all complex roots of the system e g bezout or bernshtein bound delta is the number of real roots or the roots lying in the box or disc and epsilon 2 b is the required upper bound on the output errors for computing the normal form modulo the ideal the efficient practical algorithms of b mourrain and p trebuchet in proceedings of the international symposium on symbolic and algebraic computation acm new york 2000 pp 231 238 or j c faugere j pure appl algebra 139 1999 pp 61 88 can be applied we also yield the bound oc csttn d 2 log d on the complexity of counting the numbers of all roots in a fixed box disc and all real roots for a large class of inputs and typically in practical computations the factor delta is much smaller than d delta o d this improves by the order of magnitude the known complexity estimates of the order of at least 3n d4 d3 log b or d4 which so far are the record estimates even for the approximation of a single root of a system and for each of the cited counting problems respectively our progress relies on proposing several novel techniques in particular we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators dual space of linear forms and normal forms of polynomials in the quotient algebra furthermore our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree

paraunitary filter banks over finite fields in real and complex fields unitary and paraunitary pu matrices have found many applications in signal processing there has been interest in extending these ideas to the case of finite fields we study the theory of pu filter banks fbs in gf q with q prime various properties of unitary and pu matrices in finite fields are studied in particular a number of factorization theorems are given we show that i all unitary matrices in gf q are factorizable in terms of householder like matrices and permutation matrices and ii the class of first order pu matrices the lapped orthogonal transform in finite fields can always be expressed as a product of degree one or degree two building blocks if q 2 we do not need degree two building blocks while many properties of pu matrices in finite fields are similar to those of pu matrices in complex field there are a number of differences for example unlike the conventional pu systems in finite fields there are pu systems that are unfactorizable in terms of smaller building blocks in fact in the special case of 2 spl times 2 systems all pu matrices that are factorizable in terms of degree one building blocks are diagonal matrices we derive results for both the cases of gf 2 and gf q with q 2 even though they share some similarities there are many differences between these two cases

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

backward stochastic dynamics on a filtered probability space we demonstrate that backward stochastic differential equations bsde may be reformulated as ordinary functional differential equations on certain path spaces in this framework neither ito s integrals nor martingale representation formulate are needed this approach provides new tools for the study of bsde and is particularly useful for the study of bsde with partial information the approach allows us to study the following type of backward stochastic differential equations with y t on a general filtered probability space f f t p where b is a d dimensional brownian motion l is a prescribed nonlinear mapping which sends a square integrable m to an adapted process l m and m a correction term is a square integrable martingale to be determined under certain technical conditions we prove that the system admits a unique solution y m in general the associated partial differential equations are not only nonlinear but also may be nonlocal and involve integral operators

dynamic mean variance portfolio selection with no shorting constraints this paper is concerned with mean variance portfolio selection problems in continuous time under the constraint that short selling of stocks is prohibited the problem is formulated as a stochastic optimal linear quadratic lq control problem however this lq problem is not a conventional one in that the control portfolio is constrained to take nonnegative values due to the no shorting restriction and thereby the usual riccati equation approach involving a completion of squares does not apply directly in addition the corresponding hamilton jacobi bellman hjb equation inherently has no smooth solution to tackle these difficulties a continuous function is constructed via two riccati equations and then it is shown that this function is a viscosity solution to the hjb equation solving these riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean variance problem an example illustrating these results is also presented

stochastic linear quadratic control via primal dual semidefinite programming we study stochastic linear quadratic lq optimal control problems over an infinite time horizon allowing the cost matrices to be indefinite we develop a systematic approach based on semidefinite programming sdp a central issue is the stability of the feedback control we show that this can be effectively examined through the complementary duality of the sdp furthermore we establish several implication relations among the sdp complementary duality the generalized riccati equation and the optimality of the lq control problem based on these relations we propose a numerical procedure that provides a thorough treatment of the lq control problem via primal dual sdp it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution for the latter case we develop an epsilon approximation scheme that is asymptotically optimal

potential operators associated with jacobi and fourier bessel expansions abstract we study potential operators riesz and bessel potentials associated with classical jacobi and fourier bessel expansions we prove sharp estimates for the corresponding potential kernels then we characterize those 1 p q for which the potential operators are of strong type p q of weak type p q and of restricted weak type p q these results may be thought of as analogues of the celebrated hardy littlewood sobolev fractional integration theorem in the jacobi and fourier bessel settings as an ingredient of our line of reasoning we also obtain sharp estimates of the poisson kernel related to fourier bessel expansions

well posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon abstract this paper is concerned with a stochastic linear quadratic lq problem in an infinite time horizon with multiplicative noises both in the state and the control a distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite a new type of algebraic riccati equation called a generalized algebraic riccati equation gare is introduced which involves a matrix pseudo inverse and two additional algebraic equality inequality constraints it is then shown that the well posedness of the indefinite lq problem is equivalent to a linear matrix inequality lmi condition whereas the attainability of the lq problem is equivalent to the existence of a stabilizing solution to the gare moreover all possible optimal controls are identified via the solution to the gare finally it is proved that the solution to the gare can be obtained via solving a convex optimization problem called semidefinite programming

infinite horizon and ergodic optimal quadratic control for an affine equation with stochastic coefficients we study stochastic optimal control problems with a quadratic cost and linear state equation that involves stochastic coefficients and control dependent noise and moreover is perturbed by an affine term both the infinite horizon case and the ergodic case are treated to this purpose we introduce a backward stochastic riccati equation and a dual backward stochastic equation both considered in the whole time line besides some stabilizability conditions we prove the existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems this allows us to perform the synthesis of the optimal control

on the worst case arithmetic complexity of approximating zeros of systems of polynomials let d 1 cdots d n be positive integers let mathcal p denote the set of systems of polynomials f mathbb c n to mathbb c n that have only finitely many zeros including those at infinity and that satisfy degree f i d i for all i let 0 varepsilon leqq r it is shown for fixed d 1 cdots d n that with respect to a certain model of computation the worst case computational complexity of obtaining varepsilon approximations to at least those zeros xi satisfying xi leqq r for arbitrary f in mathcal p is theta log log r varepsilon that is to say both upper and lower bounds are proved an algorithm for proving the upper bound is introduced the number of operations required by this algorithm is o left n mathcal d 4 log mathcal d log log r varepsilon n 2 mathcal d 4 left begin array 20 c 1 sigma d i n end array right 4 right quad text where mathcal d prod limits i 1

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

new optimal z complementary code sets based on generalized paraunitary matrices the concept of paraunitary pu matrices arose in the early 1990 s in the study of multi rate filter banks these matrices have found wide applications in cryptography digital signal processing and wireless communications existing pu matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice motivated by this for the first time we introduce a novel concept called z paraunitary zpu matrix whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value we show that there exists an equivalence between a zpu matrix and a z complementary code set when the latter is expressed as a matrix with polynomial entries furthermore we investigate some important properties of zpu matrices which are useful for the extension of matrix sizes and sequence lengths we propose a new construction framework for optimal zpu matrices which includes existing pu matrices as special cases finally we give the enumeration formula for the constructed z complementary sequences from the optimal zpu matrices

characterizing all optimal controls for an indefinite stochastic linear quadratic control problem this paper is concerned with a stochastic linear quadratic lq control problem in the infinite time horizon with indefinite state and control weighting matrices in the cost function it is shown that the solvability of this problem is equivalent to the existence of a so called static stabilizing solution to a generalized algebraic riccati equation moreover another algebraic riccati equation is introduced and all the possible optimal controls including the ones in state feedback form of the underlying lq problem are explicitly obtained in terms of the two riccati equations

brief paper gradient dynamic optimization with legendre chaos the polynomial chaos approach for stochastic simulation is applied to trajectory optimization by conceptually replacing random variables with free variables using the gradient method we generate with low computational cost an accurate parametrization of optimal trajectories

on stochastic riccati equations for the stochastic lqr problem abstract this paper studies an approximation of stochastic riccati equations for stochastic lqr problems some of which may be even with indefinite control weight costs

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

a block cipher cryptosystem using wavelet transforms over finite fields we present a new private key cryptosystem based on the finite field wavelet the encryption and decryption are performed by the synthesis and analysis banks of the nonlinear finite field wavelet transform whose filter coefficients are determined by the keys of the users we exploit the polyphase representation of the wavelets to introduce a shared key mechanism for the wavelet cryptosystem we propose to use wavelets that operate over gf 256 and a nonlinear device that performs a mapping on the field elements to their inverse in the field the introduced cryptographic system can operate in either stream cipher or block cipher modes depending on whether the filterbanks perform linear or circular convolution the block cipher system has a key length of 16 symbols 128 bits and an input block size of 30 symbols 240 bits to evaluate the efficiency of the developed two round wavelet cryptographic scheme we compare it with des and aes our results suggest that the wavelet cryptosystem has comparable computational complexity to aes and approximately half the complexity of des the security is tied to the length of the wavelet basis function and to the nonlinearity within the wavelet transform we study the security of the block cipher wavelet cryptosystem in response to classical attacks and those specific to this algorithm particularly those which use variations of the divide and conquer interpolation attack and discrete fourier transform techniques we show that chosen ciphertext attacks of the wavelet encryption system can reduce to the problem of solving a set of nonlinear equations over finite fields by considering existing classical and structure specific attacks we conclude that the lowest complexity of any of these attacks is greater than an exhaustive key search

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

public key cryptography using paraunitary matrices in this paper we propose an algebraic approach for designing multivariate cryptosystems our method is based on formulating a general system of multivariate polynomial equations by paraunitary matrices these matrices are a special family of invertible polynomial matrices that can be completely parameterized and efficiently generated by primitive building blocks using the general formulation that involves paraunitary matrices we design a one way function that operates over the fields of characteristic two in order to include a trapdoor we make some approximations to the paraunitary matrix the result is a trapdoor one way function that is efficient to evaluate but hard to invert unless secret information about the trapdoor is known using this function we propose a paraunitary asymmetric cryptosystem pac we present an instance of the pac and show how it can be efficiently implemented this instance operates on the finite field gf 256 the message block consists of 16 to 32 symbols from gf 256 i e the block size n is an integer between 16 and 32 the ciphertext block takes its elements from the same field and has at least ten extra symbols we show that the encryption and decryption can be efficiently performed with complexities o n3 and o n2 respectively where n is the size of the message block comparing complexities of the pac to those in the hidden field equation hfe family we show that the pac is faster in public key generation and decryption we study the computational security of the pac in addition we show that the attacks developed for the hfe are not applicable on the pac

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

holomorphic sobolev spaces associated to compact symmetric spaces using gutzmer s formula due to lassalle we characterise the images of soblolev spaces under the segal bargmann transform on compact riemannian symmetric spaces we also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the segal bargmann transform

stochastic linear quadratic control via semidefinite programming we study stochastic linear quadratic lq optimal control problems over an infinite time horizon allowing the cost matrices to be indefinite we develop a systematic approach based on semidefinite programming sdp a central issue is the stability of the feedback control and we show this can be effectively examined through the complementary duality of the sdp furthermore we establish several implication relations among the sdp complementary duality the generalized riccati equation and the optimality of the lq control problem based on these relations we propose a numerical procedure that provides a thorough treatment of the lq control problem via primal dual sdp it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution for the latter case we develop an epsilon approximation scheme that is asymptotically optimal

stochastic control for linear systems driven by fractional noises this paper is concerned with optimal control of stochastic linear systems involving fractional brownian motion fbm first as a prerequisite for studying the underlying control problems some new results on stochastic integrals and stochastic differential equations associated with fbm are established then three control models are formulated and studied in the first two models the state is scalar valued and the control is taken as markovian either the problems are completely solved based on a riccati equation for model 1 where the cost is a quadratic functional on state and control variables or optimality is characterized for model 2 where the cost is a power functional the last control model under investigation is a general one where the system involves the stratonovich integral with respect to fbm the state is multidimensional and the admissible controls are not limited to being markovian a new riccati type equation which is a backward stochastic differential equation involving both fbm and normal brownian motion is introduced optimal control and optimal value of the model are explicitly obtained based on the solution to this riccati type equation

relationship between backward stochastic differential equations and stochastic controls a linear quadratic approach it is well known that backward stochastic differential equations bsdes stem from the study on the pontryagin type maximum principle for optimal stochastic controls a solution of a bsde hits a given terminal value which is a random variable by virtue of an it additional martingale term and an indefinite initial state this paper attempts to explore the relationship between bsdes and stochastic controls by interpreting bsdes as some stochastic optimal control problems more specifically associated with a bsde a new stochastic control problem is introduced with the same dynamics but a definite given initial state the martingale term in the original bsde is regarded as the control and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the bsde this problem is solved in a closed form by the stochastic linear quadratic lq theory developed recently the general result is then applied to the black scholes model where an optimal mean variance hedging portfolio is obtained explicitly in terms of the option price finally a modified model is investigated where the difference between the state and the expectation of the given terminal value at any time is taken into account

constrained stochastic lq control with random coefficients and application to portfolio selection this paper is devoted to the study of a stochastic linear quadratic lq optimal control problem where the control variable is constrained in a cone and all the coefficients of the problem are random processes employing tanaka s formula optimal control and optimal cost are explicitly obtained via solutions to two extended stochastic riccati equations esres the esres introduced for the first time in this paper are highly nonlinear backward stochastic differential equations bsdes whose solvability is proved based on a truncation function technique and kobylanski s results the general results obtained are then applied to a mean variance portfolio selection problem for a financial market with random appreciation and volatility rates and with short selling prohibited feasibility of the problem is characterized and efficient portfolios and efficient frontier are presented in closed forms

error estimates for the logarithmic barrier method in linear quadratic stochastic optimal control problems we consider a linear quadratic stochastic optimal control problem whith non negativity control constraints the latter are penalized with the classical logarithmic barrier using a duality argument and the stochastic minimum principle we provide an error estimate for the solution of the penalized problem which is the natural extension of the well known estimate in the deterministic framework

mean variance portfolio selection with random parameters in a complete market this paper concerns the continuous time mean variance portfolio selection problem in a complete market with random interest rate appreciation rates and volatility coef cients the problem is tackled using the results of stochastic linear quadratic lq optimal control and backward stochastic differential equations bsdes two theories that have been extensively studied and developed in recent years specifically the mean variance problem is formulated as a linearly constrained stochastic lq control problem solvability of this lq problem is reduced in turn to proving global solvability of a stochastic riccati equation the proof of existence and uniqueness of this riccati equation which is a fully nonlinear and singular bsde with random coefficients is interesting in its own right and relies heavily on the structural properties of the equation efficient investment strategies as well as the mean variance efficient frontier are then analytically derived in terms of the solution of this equation in particular it is demonstrated that the efficient frontier in the mean standard deviation diagram is still a straight line or equivalently risk free investment is still possible even when the interest rate is random finally a version of the mutual fund theorem is presented

indefinite stochastic linear quadratic control and generalized differential riccati equation a stochastic linear quadratic lq control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite indefinite stochastic lq theory has been extensively developed and has found interesting applications in finance however there remains an outstanding open problem which is to identify an appropriate riccati type equation whose solvability is it equivalent to the solvability of the indefinite stochastic lq problem this paper solves this open problem for lq control in a finite time horizon a new type of differential riccati equation called the generalized differential riccati equation is introduced which involves algebraic equality inequality constraints and a matrix pseudoinverse it is then shown that the solvability of the generalized riccati equation is not only sufficient but also necessary for the well posedness of the indefinite lq problem and the existence of optimal feedback open loop controls moreover all of the optimal controls can be identified via the solution to the riccati equation an example is presented to illustrate the theory developed

on the backward stochastic riccati equation in infinite dimensions we study backward stochastic riccati equations bsres arising in quadratic optimal control problems with infinite dimensional stochastic differential state equations we allow the coefficients both in the state equation and in the cost to be random in such a context bsres are backward stochastic differential equations existing in a non hilbert space and involving quadratic nonlinearities we propose two different notions of solutions to bsres and prove for both of them existence and uniqueness results we also show that such solutions allow us to perform the synthesis of the optimal control finally we apply our results to the optimal control of a delay equation and of a wave equation with random damping

minimization of risk and linear quadratic optimal control theory this article is concerned with the optimal control problem for the linear stochastic system xt x t 0 asxs bsus fs ds t 0 d i 1 ci s xs di s us gi s dwi s with the convex risk functional j u em xt et 0 g t xt u t dt in order to guarantee the existence of an optimal control without any weak compactness assumption on the admissible control set we assume that the risk function m is coercive and thatd 1 d di is uniformlypositive rather than to assume like in the control literature that the running risk function g is coercive with respect to the control variable in this new setting the running risk function g maybe independent of the control variable and therefore the so called singular linear quadratic lq stochastic control problem is included a rigorous theoryis developed for the general stochastic lq problem with random coefficients and the bounded mean oscillation martingale theoryis used to account for the concerned integrability it plays a crucial role in the following exposition a to connect the stochastic lq problem to two associated backward stochastic differential equations bsdes one is an n n symmetric matrix valued nonlinear riccati bsde and the other is an n dimensional linear bsde with unbounded coefficients b to show that the latter bsde has an adapted solution pair of the suitablynecessaryregularity this seems to be the first application in a stochastic lq theory of the bmo martingale theory which roots in harmonic analysis furthermore with the help of an a priori estimate on the risk functional existence and uniqueness of the solutions of backward stochastic riccati differential equations bsrdes in the singular case is reduced to the regular case via a perturbation method and then a new existence and uniqueness result on bsrdes is obtained for the singular case

two band wavelets and filterbanks over finite fields with connections to error control coding recently we have developed a new framework to study error control coding using finite field wavelets and filterbanks fbs this framework reveals a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones the paper introduces the theory of wavelet decompositions of signals in vector spaces defined over galois fields to avoid the limitations of the number theoretic fourier transform our wavelet transform relies on a basis decomposition in the time rather than the frequency domain first by employing a symmetric nondegenerate canonical bilinear form we obtain a necessary and sufficient condition that the basis functions defined over finite fields must satisfy in order to construct an orthogonal wavelet transform then we present a design methodology to generate the mother wavelet and scaling function over finite fields by relating the wavelet transform to two channel paraunitary pu fbs finally we describe the application of this transform to the construction of error correcting codes in particular we give examples of double circulant codes that are generated by wavelets

theory of paraunitary filter banks over fields of characteristic two motivated by our wavelet framework for error control coding we proceed to develop an important family of wavelet transforms over finite fields paraunitary pu filter banks that are realizations of orthogonal wavelets by multirate filters are an important subclass of perfect reconstruction pr filter banks a parameterization of pu filter banks that covers all possible pu systems is very desirable in error control coding because it provides a framework for optimizing the free parameters to maximize coding performance this paper undertakes the problem of classifying all pu matrices with entries from a polynomial ring where the coefficients of the polynomials are taken from finite fields it constructs householder transformations that are used as elementary operations for the realization of all unitary matrices then it introduces elementary pu building blocks and a factorization technique that is specialized to obtain a complete realization for all pu filter banks over fields of characteristic two this is proved for the 2 spl times 2 case and conjectured for the m spl times m case where m spl ges 3 using these elementary building blocks we can construct all pu filter banks over fields of characteristic two these filter banks can be used to implement transforms which in turn provide a powerful new perspective on the problems of constructing and decoding arbitrary rate error correcting codes

fractional integrals on compact riemannian symmetric spaces of rank one abstract in this paper we study mixed norm boundedness for fractional integrals related to laplace beltrami operators on compact riemannian symmetric spaces of rank one the key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric jacobi polynomials expansions in particular we find a novel sharp estimate for the jacobi fractional integral kernel with explicit dependence on the type parameters

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

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

a local paley wiener theorem for compact symmetric spaces abstract the fourier coefficients of a smooth k invariant function on a compact symmetric space m u k are given by integration of the function against the spherical functions for functions with support in a neighborhood of the origin we describe the size of the support by means of the exponential type of a holomorphic extension of the fourier coefficients

a general maximum principle for optimal control of forward backward stochastic systems abstract a general maximum principle for optimal control problems derived by forward backward stochastic systems is established where control domains are non convex and forward diffusion coefficients explicitly depend on control variables these optimal control problems have broad applications in mathematical finance and economics such as the recursive mean variance portfolio choice problems the maximum principle is applied to study a forward backward linear quadratic optimal control problem with a non convex control domain an optimal solution is obtained

the risk sensitive maximum principle for controlled forward backward stochastic differential equations abstract in this paper we consider the risk sensitive optimal control problem for forward backward stochastic differential equations fbsdes we consider two different cases case 1 the drift and diffusion terms of the forward part are independent of the backward part not fully coupled fbsde and the running cost does not include the backward part case 2 the fully coupled fbsde with terminal and initial costs in both cases the risk sensitive objective functional is considered the corresponding control domain is not necessarily convex and the diffusion term of the forward part depends on control and state variables in both cases by using the nonlinear transformations of the equivalent risk neutral problems we obtain two different risk sensitive maximum principles which are characterized in terms of the variational inequalities we show that the risk sensitive maximum principle of case 1 consists of the two coupled adjoint equations with an additional scalar adjoint equation which are backward stochastic differential equations bsdes the risk sensitive maximum principle of case 2 consists of the two coupled adjoint equations which are coupled fbsdes with an additional scalar adjoint equation and optimality conditions in both cases the additional scalar adjoint equations are bsdes which are induced due to the nonlinear transformations of the adjoint processes in the equivalent risk neutral problems through the application of results we consider the linear quadratic risk sensitive optimal control problem for the linear fbsde and by using the risk sensitive maximum principle of case 1 we characterize an explicit optimal solution

quadratic hedging and mean variance portfolio selection with random parameters in an incomplete market this paper concerns the problems ofquadratic hedging andpricing andmean variance portfolio selection in anincomplete market setting with continuous trading multiple assets and brownian information in particular we assume throughout that the parameters describing the market model may be random processes we approach these problems from the perspective oflinear quadratic lq optimal control and backward stochastic differential equations bsdes that is we focus on the so calledstochastic riccati equation sre associated with the problem excepting certain special cases solvability of the sre remains an open question our primary theoretical contribution is a proof of existence and uniqueness of solutions of the sre associated with the quadratic hedging and mean variance problems in addition we derive closed form expressions for the optimal portfolios and efficient frontier in terms of the solution of the sre a generalization of themutual fund theorem is also obtained

solvability and asymptotic behavior of generalized riccati equations arising in indefinite stochastic lq controls the optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential riccati equation in this paper we provide general necessary and sufficient conditions for the solvability of this generalized differential riccati equation furthermore its asymptotic behavior is investigated along with its connection to the generalized algebraic riccati equation associated with the linear quadratic control problem in finite time horizon examples are presented to illustrate the results established

mean variance hedging and stochastic control beyond the brownian setting we show for continuous semimartingales in a general filtration how the mean variance hedging problem can be treated as a linear quadratic stochastic control problem the adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process and we give necessary and sufficient conditions for the solvability of these generalized stochastic riccati equations motivated from mathematical finance this paper takes a first step toward linear quadratic stochastic control in more general than brownian settings

sobolev spaces associated with jacobi expansions abstract we define and study sobolev spaces associated with jacobi expansions we prove that these sobolev spaces are isomorphic to jacobi potential spaces as a technical tool we also show some approximation properties of poisson jacobi integrals