mean field game for linear quadratic stochastic recursive systems abstract this paper focuses on linear quadratic lq for short mean field games described by forward backward stochastic differential equations fbsdes for short in which the individual control region is postulated to be convex the decentralized strategies and consistency condition are represented by a kind of coupled mean field fbsdes with projection operators the well posedness of consistency condition system is obtained using the monotonicity condition method the nash equilibrium property is discussed as well

constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

harmonic analysis of stochastic equations and backward stochastic differential equations the bmo martingale theory is extensively used to study nonlinear multi dimensional stochastic equations in mathcal r p p in 1 infty and backward stochastic differential equations bsdes in mathcal r p times mathcal h p p in 1 infty and in mathcal r infty times overline l infty rm bmo with the coefficients being allowed to be unbounded in particular the probabilistic version of fefferman s inequality plays a crucial role in the development of our theory which seems to be new several new results are consequently obtained the particular multi dimensional linear cases for stochastic differential equations sdes and bsdes are separately investigated and the existence and uniqueness of a solution is connected to the property that the elementary solutions matrix for the associated homogeneous sde satisfies the reverse holder inequality for some suitable exponent p 1 finally some relations are established between kazamaki s quadratic critical exponent b m of a bmo martingale m and the spectral radius of the stochastic integral operator with respect to m which lead to a characterization of kazamaki s quadratic critical exponent of bmo martingales being infinite

multidimensional probability density function approximations for detection classification and model order selection this paper addresses the problem of calculating the multidimensional probability density functions pdfs of statistics derived from known many to one transformations of independent random variables rvs with known distributions the statistics covered in the paper include reflection coefficients autocorrelation estimates cepstral coefficients and general linear functions of independent rvs through pdf transformation these results can be used for general pdf approximation detection classification and model order selection a model order selection example that shows significantly better performance than the akaike and mdl method is included

stochastic linear quadratic control with conic control constraints on an infinite time horizon this paper is concerned with a stochastic linear quadratic lq control problem in the infinite time horizon where the control is constrained in a given arbitrary closed cone the cost weighting matrices are allowed to be indefinite and the state is scalar valued first the mean square conic stabilizability of the system is defined which is then characterized by a set of simple conditions involving linear matrix inequalities lmis next the issue of well posedness of the underlying optimal lq control which is necessitated by the indefiniteness of the problem is addressed in great detail and necessary and sufficient conditions of the well posedness are presented on the other hand to address the lq optimality two new algebraic equations a la riccati called extended algebraic riccati equations eares along with the notion of their stabilizing solutions are introduced for the first time optimal feedback control as well as the optimal value are explicitly derived in terms of the stabilizing solutions to the eares moreover several cases when the stabilizing solutions do exist are discussed and algorithms of computing the solutions are presented finally numerical examples are provided to illustrate the theoretical results established

separated continuous conic programming strong duality and an approximation algorithm motivated by recent applications in robust optimization and in sign constrained linear quadratic control we study in this paper a new class of optimization problems called separated continuous conic programming sccp focusing on a symmetric primal dual pair we develop a strong duality theory for the sccp our idea is to use discretization to connect the sccp and its dual to two ordinary conic programs we show if the latter are strongly feasible and with finite optimal values a condition that is readily verifiable then the strong duality holds for the sccp this approach also leads to a polynomial time approximation algorithm that solves the sccp to any required accuracy

uniform manifold sampling ums sampling the maximum entropy pdf maximum entropy pdf projection mepp is a way to construct generative models from feature transformations corresponding to each dimension reducing feature mapping such as a feed forward neural network or an algorithm to calculate linear prediction coefficients from time series and given a prior distribution for the features there exists a unique generative model for the input data which subject to mild requirements is maximum entropy maxent among all probability density functions pdfs that are consistent with the given feature prior in this paper we consider the problem of sampling from these maxent projected pdfs the sampling process consists of drawing a sample from the given feature prior distribution then drawing samples uniformly distributed on the inversion set set of input samples consistent with the drawn feature value usually a manifold the process is called uniform manifold sampling ums we describe ums for simple nonlinear and iterative feature transformations then focus on linear transformations with input data constraints x i 0 or 0 leq x i leq 1 which require mcmc based sampling we discover that the manifold centroid sample mean for a fixed feature value is useful as a deterministic maxent feature inversion solution we show how to predict the centroid efficiently without sampling and demonstrate its usefulness in speeding up mcmc by an order of magnitude and in spectral estimation and image reconstruction finally we provide an example of ums in a classification experiment in which we use monte carlo integration to create true generative models from arbitrary classifiers

non representable hyperbolic matroids the generalized lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices hyperbolic polynomials give rise to a class of hyperbolic matroids which properly contains the class of matroids representable over the complex numbers this connection was used by the first author to construct counterexamples to algebraic stronger versions of the generalized lax conjecture by considering a non representable hyperbolic matroid the va mos matroid and a generalization of it are to this day the only known instances of non representable hyperbolic matroids we prove that the non pappus and non desargues matroids are non representable hyperbolic matroids by exploiting a connection due to jordan between euclidean jordan algebras and projective geometries we further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non representable finally we explore consequences to algebraic versions of the generalized lax conjecture

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

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

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

on the positivity of symmetric polynomial functions part i general results abstract we prove that a real symmetric polynomial inequality of degree d 2 holds on r n if and only if it holds for elements with at most d 2 distinct non zero components which may have multiplicities we establish this result by solving a cauchy problem for ordinary differential equations involving the symmetric power sums this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function in the final section extensions of our results to the whole space r n are outlined the main results are theorems 5 1 and 5 2 with corollaries 2 1 and 5 2 and the corresponding results for r n from the last subsection part ii will contain a discussion on the ordered vector space h d n in general as well as on the particular cases of degrees d 4 and d 5 finite test sets for positivity in the homogeneous case and other sufficient criteria

mean variance hedging when there are jumps in this paper we consider the problem of mean variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by brownian motion and doubly stochastic poisson processes this problem is formulated as a stochastic control problem and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations the results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modeled by the arrivals of poisson processes with stochastic intensities applications to the problem of hedging default risk are also discussed

on positive controllers and linear quadratic optimal control problems an assumption on the non negative solutions of quadratic equations has been used to study linear quadratic optimal control problems with positive controllers this note investigates this assumption in more detail

quantum entanglement symmetric nonnegative quadratic polynomials and moment problems quantum states are represented by positive semidefinite hermitian operators with unit trace known as density matrices an important subset of quantum states is that of separable states the complement of which is the subset of entangled states we show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis only a small number of such moments are accessible experimentally and so in practice the question of quantum entanglement of a many body system e g a system consisting of several atoms can be reduced to a truncated moment problem by considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in mathbb r d we work with moments up to degree 2 only since these are most readily available experimentally we derive necessary and sufficient conditions for belonging to the moment cone which can be expressed via a linear matrix inequality of size at most 2d 2 which is independent of n the linear matrix inequalities can be converted into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone and show that the two conditions approach each other in the limit of large n the inequalities are derived via considering the dual cone of nonnegative polynomials and its sum of squares relaxation we show that the sum of squares relaxation of the dual cone is asymptotically exact and using symmetry reduction techniques blekherman and riener symmetric nonnegative forms and sums of squares arxiv 1205 3102 2012 gatermann and parrilo j pure appl algebra 192 1 3 95 128 https doi org 10 1016 j jpaa 2003 12 011 2004 it can be written as a small linear matrix inequality of size at most 2d 2 which is independent of n for the cone of symmetric nonnegative polynomials with the relevant support we also prove an analogue of the half degree principle for globally nonnegative symmetric polynomials riener j pure appl algebra 216 4 850 856 https doi org 10 1016 j jpaa 2011 08 012 2012 timofte j math anal appl 284 1 174 190 https doi org 10 1016 s0022 247x 03 00301 9 2003

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

on the positivity of symmetric polynomial functions part ii lattice general results and positivity criteria for degrees 4 and 5 abstract we prove that homogeneous symmetric polynomial inequalities of degree p 4 5 in n positive 1 variables can be algorithmically tested on a finite set depending on the given inequality theorem 13 the test set can be obtained by solving a finite number of equations of degree not exceeding p 2 section 3 discusses the structure of the ordered vector spaces h p n and h p n in section 4 positivity criteria for degrees 4 and 5 are stated and proved the main results are theorems 10 14 part iii of this work will be concerned with the construction of extremal homogeneous symmetric polynomials best inequalities of degree 4 in n positive variables

explicit solution for constrained scalar state stochastic linear quadratic control with multiplicative noise we study in this paper a class of constrained linear quadratic lq optimal control problem formulations for the scalar state stochastic system with multiplicative noise which has various applications especially in the financial risk management the linear constraint on both the control and state variables considered in our model destroys the elegant structure of the conventional lq formulation and has blocked the derivation of an explicit control policy so far in the literature we successfully derive in this paper the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure we reveal that the optimal control policy is a piecewise affine function of the state and can be computed offline efficiently by solving two coupled riccati equations under some mild conditions we also obtain the stationary control policy for an infinite time horizon we demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems

discriminants of cyclic homogeneous inequalities of three variables abstract in this article we study the structure of the cone of semidefinite forms it is a closed semialgebraic set but usually is not basic closed semialgebraic set a discriminant is a defining equation of an irreducible component of algebraic boundary of this cone we calculate discriminants using new tools characteristic variety and local cones a characteristic variety is a semialgebraic subset of a real projective variety on which the family of inequalities is essentially defined as linear functions local cone is a subcone of the psd cone which corresponds to a maximal ideal this theory works well for a family of polynomials which are invariant under an action of a finite group after we construct an abstract general theory we apply it to a family of cyclic homogeneous polynomials of three real variables of degree d we calculate some discriminants for d 3 4 5 and 6 and we show that this theory derives many new results

maximum entropy pdf design using feature density constraints applications in signal processing this paper revisits an existing method of constructing high dimensional probability density functions pdfs based on the pdf at the output of a dimension reducing feature transformation we show how to modify the method so that it can provide the pdf with the highest entropy among all pdfs that generate the given low dimensional pdf the method is completely general and applies to arbitrary feature transformations the chain rule is described for multi stage feature calculations typically used in signal processing examples are given including mfcc and auto regressive features experimental verification of the results using simulated data is provided including a comparison with competing generative methods

time inconsistent stochastic linear quadratic control in this paper we formulate a general time inconsistent stochastic linear quadratic lq control problem the time inconsistency arises from the presence of a quadratic term of the expected state as well as a state dependent term in the objective functional we define an equilibrium instead of optimal solution within the class of open loop controls and derive a sufficient condition for equilibrium controls via a flow of forward backward stochastic differential equations when the state is one dimensional and the coefficients in the problem are all deterministic we find an explicit equilibrium control as an application we then consider a mean variance portfolio selection model in a complete financial market where the risk free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes applying the general sufficient condition we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic

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

on the positivity of symmetric polynomial functions part iii extremal polynomials of degree 4 abstract in this paper which is a continuation of v timofte on the positivity of symmetric polynomial functions part i general results j math anal appl 284 2003 174 190 and v timofte on the positivity of symmetric polynomial functions part ii lattice general results and positivity criteria for degrees 4 and 5 j math anal appl in press we study properties of extremal polynomials of degree 4 and we give the construction of some of them the main results are theorems 9 13 15 16 and 18

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

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

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

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

on continuous time constrained stochastic linear quadratic control this paper studies a class of continuous time scalar state stochastic linear quadratic lq optimal control problems with the linear control constraints using the state separation theorem induced from its special structure we derive the analytical solution for this class of problems the revealed optimal control policy is a piece wise affine function of system state this control policy can be computed efficiently by solving two riccati equations off line under some mild conditions the stationary optimal control policy can be also achieved for this class of problems over an infinite horizon examples are presented to shed light on the theoretical results established

asymptotically optimal approximation of multidimensional pdf s by lower dimensional pdf s probability density functions pdf s of high dimensionality are impractical to estimate from real data for accurate estimation the dimensionality of the pdf can be at most 5 10 in order to reduce the dimensionality a sufficient statistic is usually employed when none is available there is no universal agreement on how to proceed we show how to construct a high dimension pdf based on the pdf of a low dimensional statistic that is closest to the true one in the sense of divergence the latter criterion asymptotically minimizes the probability of error in a decision rule an application to feature selection for classification is described

the pdf projection theorem and the class specific method we present the theoretical foundation for optimal classification using class specific features and provide examples of its use a new probability density function pdf projection theorem makes it possible to project probability density functions from a low dimensional feature space back to the raw data space an m ary classifier is constructed by estimating the pdfs of class specific features then transforming each pdf back to the raw data space where they can be fairly compared although statistical sufficiency is not a requirement the classifier thus constructed becomes equivalent to the optimal bayes classifier if the features meet sufficiency requirements individually for each class this classifier is completely modular and avoids the dimensionality curse associated with large complex problems by recursive application of the projection theorem it is possible to analyze complex signal processing chains we apply the method to feature sets including linear functions of independent random variables cepstrum and mel cepstrum in addition we demonstrate how it is possible to automate the feature and model selection process by direct comparison of log likelihood values on the common raw data domain

sufficiency classification and the class specific feature theorem a new proof of the class specific feature theorem is given the proof makes use of the observed data as opposed to the set of sufficient statistics as in the original formulation we prove the theorem for the classical case in which the parameter vector is deterministic and known as well as for the bayesian case in which the parameter vector is modeled as a random vector with known prior probability density function the essence of the theorem is that with a suitable normalization the probability density function of the sufficient statistic for each probability density function family can be used for optimal classification one need not have knowledge of the probability density functions of the data under each hypothesis

exactness of sums of squares relaxations involving 3x3 matrices and lorentz cones we consider a family pm n of cones of positive maps and a semidefinite relaxation of these cones the cone pm n can be described as the set of those linear mappings from the space rm into the space of real symmetric n n matrices which map the m dimensional lorentz cone into the cone of real symmetric positive semidefinite matrices we describe the cone pm n as a cone of nonnegative polynomials in several variables we show that the considered semidefinite relaxation is in fact a sums of squares relaxation corresponding to this description of pm n our main result is that for n 3 the relaxation is exact hence it yields the exact result for optimisation problems over the cones pm 3 in particular the matrix ellipsoid problem for real symmetric 3 3 matrices can be rewritten as feasibility problem of a linear matrix inequality for m 4 n 4 there exist points in pm n which do not lie in the semidefinite set corresponding to the relaxation hence the relaxation is exact if and only if min n m 3

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

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

linear quadratic regulator problem with positive controls in this paper the linear quadratic regulator problem with a positivity constraint on the admissible control set is addressed necessary and sufficient conditions for optimality are presented in terms of inner products projections on closed convex sets pontryagin s maximum principle and dynamic programming the main results are concerned with smoothness of the optimal control and the value function the maximum principle will be extended to the infinite horizon case based on these analytical methods we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem the numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other

real even symmetric ternary forms let sen m denote the set of all real symmetric forms of degree m 2d let psen m and sen m denote the cones of positive semidefinite psd and sum of squares sos elements of sen m respectively for m 2 or 4 these cones coincide for m 6 they do not and were analyzed in even symmetric sextics by m d choi t y lam and b reznick 1987 math z 195 pp 559 580 we present an easily checked necessary and sufficient condition for an even symmetric n ary octic to be in psen 8 and for an even symmetric ternary decic to be in pse3 10 we also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10 we proceed to discuss the extremal elements of the cones pse3 8 and pse3 10 this leads to the question how many of these extremal forms have sos representations we prove that pse3 8 se3 8 and demonstrate that pse3 10 se3 10 is nonempty providing many new examples of psd forms which are not sos we give a graphic representation of ternary forms which also indicates whether or not an element of se3 8 or se3 10 is psd we interpret elements of pse3 m as inequalities in particular we give all symmetric polynomial inequalities of degree 5 satisfied by the sides of a triangle

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

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

model predictive control for constrained systems with serially correlated stochastic parameters and portfolio optimization in this paper we consider mpc for constrained discrete time systems with stochastic parameters which are assumed to be a set of serially correlated time series a generalized performance criterion is composed of a weighted sum of a linear combination of the a expected value of quadratic forms of state and control vectors b quadratic forms of the expected value of the state vector and c the linear component of the expected value of the state vector the purpose of the present paper is to design optimal control strategies that are independent of distributional assumptions on the stochastic parameters and subject to hard constraints on the input manipulated variables and to provide a numerically tractable algorithm for practical applications all expressions are presented in terms of the first and second order conditional moments the results are applied to a problem of investment portfolio optimization with serially correlated returns we present the numerical modelling results based on stocks traded on the russian stock exchanges micex

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

numerical methods for portfolio selection with bounded constraints this work develops an approximation procedure for portfolio selection with bounded constraints based on the markov chain approximation techniques numerical procedures are constructed for the utility optimization task under simple conditions the convergence of the approximation sequences to the wealth process and the optimal utility function is established numerical examples are provided to illustrate the performance of the algorithms