linear quadratic partially observed forward backward stochastic differential games and its application in finance abstract this paper is concerned with a partially observed linear quadratic game problem driven by forward backward stochastic differential equations where the forward diffusion coefficients do not contain control variables and the control domains are not necessarily convex the drift term of the observation equation is linear with respect to the state and there is correlated noise between the state and the observation equation by virtue of the classical spike variational method and the backward separation technique we derive a necessary and a sufficient condition of the stochastic differential game problem then we obtain filtering equations and present a feedback representation form of the equilibrium point through riccati equations as a practical application we solve a partial information investment problem involving g expectation as a convex risk measurement and give the numerical simulation to show the explicit investment strategy and illustrate some reasonable phenomena influenced by common financial parameters

optimal designs for space time linear precoders and decoders we introduce a new paradigm for the design of transmitter space time coding that we refer to as linear precoding it leads to simple closed form solutions for transmission over frequency selective multiple input multiple output mimo channels which are scalable with respect to the number of antennas size of the coding block and transmit average peak power the scheme operates as a block transmission system in which vectors of symbols are encoded and modulated through a linear mapping operating jointly in the space and time dimension the specific designs target minimization of the symbol mean square error and the approximate maximization of the minimum distance between symbol hypotheses under average and peak power constraints the solutions are shown to convert the mimo channel with memory into a set of parallel flat fading subchannels regardless of the design criterion while appropriate power bits loading on the subchannels is the specific signature of the different designs the proposed designs are compared in terms of various performance measures such as information rate ber and symbol mean square error

existence conditions and variational approach for adapted solutions of the two point boundary value problem of stochastic differential equations abstract this paper considers the two point boundary value problem of stochastic differential equation with the following form dx t f t x t dt t x t dw t ax 0 bx t the sufficient and necessary conditions are given for the existence of the adapted solutions in the simple case that f t xt ft the solution can be obtained by introducing a control term ft extending the solution from xt to xt ft and constructing a process sequence all the results obtained are compared with those related to the eigenvalue problems and the martingale approach solution proposed by pardoux and peng for backward stochastic differential equations

closed formulae for a parametric mixed sensitivity problem abstract mixed sensitivity optimization is a two block problem which normally requires an iterative solution the purpose of this paper is to give an explicit closed formula for the optimal cost in the case of a parametric mixed sensitivity problem we also show that all the controllers which satisfy an arbitrary h norm bound may be parameterized in terms of the solutions to the standards h 2 riccati equations we hope that these results will help progress towards closed formulae for the optimal cost in a wider class of problems

well posedness of backward stochastic differential equations with general filtration abstract this paper is addressed to the well posedness of some linear and semilinear backward stochastic differential equations with general filtration without using the martingale representation theorem the point of our approach is to introduce a new notion of solution i e the transposition solution which coincides with the usual strong solution when the filtration is natural but it is more flexible for the case of general filtration than the existing notion of solutions a comparison theorem for transposition solutions and a pontryagin type stochastic maximum principle are also presented

on the integral representation of g expectations with terminal constraints abstract in this paper we study the integral representation of g expectations with two kinds of terminal constraints and obtain the corresponding necessary and sufficient conditions

anticipated backward doubly stochastic differential equations in this paper we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations anticipated bdsdes the coefficients of these bdsdes depend on the future value of the solution y z we obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations besides as an application we also establish a duality between the anticipated bdsdes and the delayed doubly stochastic differential equations delayed dsdes

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

design and analysis of transmit beamforming based on limited rate feedback this paper deals with design and performance analysis of transmit beamformers for multiple input multiple output mimo systems based on bandwidth limited information that is fed back from the receiver to the transmitter by casting the design of transmit beamforming based on limited rate feedback as an equivalent sphere vector quantization svq problem multiantenna beamformed transmissions through independent and identically distributed i i d rayleigh fading channels are first considered the rate distortion function of the vector source is upper bounded and the operational rate distortion performance achieved by the generalized lloyd s algorithm is lower bounded although different in nature the two bounds yield asymptotically equivalent performance analysis results the average signal to noise ratio snr performance is also quantified finally beamformer codebook designs are studied for correlated rayleigh fading channels and a low complexity codebook design that achieves near optimal performance is derived

periodic homogenization for inertial particles we study the problem of homogenization for inertial particles moving in a periodic velocity field and subject to molecular diffusion we show that under appropriate assumptions on the velocity field the large scale long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone to achieve this we use a formal multiple scale expansion in the scale parameter this expansion relies on the hypo ellipticity of the underlying diffusion an expression for the diffusivity tensor is found and various of its properties studied in particular an expansion in terms of the non dimensional particle relaxation time the stokes number is shown to co incide with the known result for passive non inertial tracers in the singular limit 0 this requires the solution of a singular perturbation problem achieved by means of a formal multiple scales expansion in incompressible and potential fields are studied as well as fields which are neither and theoretical findings are supported by numerical simulations

brief infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems this paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation like deterministic control systems it is shown that the exact controllability of an open loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed loop stochastic system formed by means of suitable linear feedback of the states as an application a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward backward stochastic differential equations is provided

bit interleaved coded multiple beamforming with imperfect csit this paper addresses the performance of bit interleaved coded multiple beamforming bicmb 1 2 with imperfect knowledge of beamforming vectors most studies for limited rate channel state information at the transmitter csit assume that the precoding matrix has an invariance property under an arbitrary unitary transform in bicmb this property does not hold on the other hand the optimum precoder and detector for bicmb are invariant under a diagonal unitary transform in order to design a limited rate csit system for bicmb we propose a new distortion measure optimum under this invariance based on this new distortion measure we introduce a new set of centroids and employ the generalized lloyd algorithm for codebook design we provide simulation results demonstrating the performance improvement achieved with the proposed distortion measure and the codebook design for various receivers with linear detectors we show that although these receivers have the same performance for perfect csit their performance varies under imperfect csit

quadratic bsde with mathbb l 2 terminal data krylov s estimate ito krylov s formula and existence results we establish a krylov type estimate and an ito krylov change of variable formula for the solutions of one dimensional quadratic backward stochastic differential equations qbsdes with a measurable generator and an arbitrary terminal datum this allows us to prove various existence and uniqueness results for some classes of qbsdes with a square integrable terminal condition and sometimes a merely measurable generator it turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and or uniqueness of solutions we also establish a comparison theorem for solutions of a particular class of qbsdes with measurable generator as a byproduct we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations qpdes with a square integrable terminal datum

parameter estimation for multiscale diffusions we study the problem of parameter estimation for time series possessing two widely separated characteristic time scales the aim is to understand situations where it is desirable to fit a homogenized single scale model to such multiscale data we demonstrate numerically and analytically that if the data is sampled too finely then the parameter fit will fail in that the correct parameters in the homogenized model are not identified we also show numerically and analytically that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly the ideas are studied in the context of thermally activated motion in a two scale potential however the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data

sub concave and sub convex capacities abstract in this study we propose the concepts of sub concave and sub convex capacities first we investigate some properties of sub concave and sub convex capacities second we discuss the choquet integrals with respect to sub concave and sub convex capacities we then show that the upper and lower probabilities comprising the supremum and infimum over the set of risk neutral martingale measures in the asymmetrical case are sub concave and sub convex capacities respectively

finding adapted solutions of forward backward stochastic differential equations method of continuation the notion of bridge is introduced for systems of coupled forward backward stochastic differential equations fbsdes for short this notion helps us to unify the method of continuation in finding adapted solutions to such fbsdes over any finite time durations it is proved that if two fbsdes are linked by a bridge then they have the same unique solvability consequently by constructing appropriate bridges we obtain several classes of uniquely solvable fbsdes

an optimal two transmit antenna space time code and its stacked extensions a space time code is proposed that exhibits the highest coding gain among competing full rate full transmit diversity space time codes for the two transmit and receive antenna coherent quasi static fading channel the proposed code is derived from a layered architecture with real rotation of quadrature amplitude modulation qam information symbols in two dimensions the existing codes of similar architecture concentrate on application of complex full modulation diversity rotations or asymmetric real rotations an analytic evaluation illustrates the significant improvement in coding gain achieved with the proposed code moreover the coding gain of the proposed code is independent of its rate this implies that the proposed code achieves the optimal diversity multiplexing tradeoff curve for the two transmit antenna system a stacked extension of the proposed code offers a reduced complexity capacity optimal alternative to the full diversity codes for larger number of transmit antennas performance enhancement in several scenarios is verified through simulations

homogenization of linear and semilinear second order parabolic pdes with periodic coefficients a probabilistic approach we study the limit of the solution of linear and semilinear second order pdes of parabolic type with rapidly oscillating periodic coefficients singular drift and singular coefficient of the zeroth order term our method of proof is fully probabilistic and builds upon the arguments in earlier work in the linear case we use the feynman kac formula to represent the solution of the parabolic pde and in the semilinear case we use an associated backward stochastic differential equation

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

quasi linear pdes and forward backward stochastic differential equations weak solutions abstract in this paper we study the existence uniqueness and the probabilistic representation of the weak solutions of quasi linear parabolic and elliptic partial differential equations pdes in the sobolev space h 1 r d for this we study first the solutions of forward backward stochastic differential equations fbsdes with smooth coefficients regularity of solutions and their connection with classical solutions of quasi linear parabolic pdes then using the approximation procedure we establish their convergence in the sobolev space to the solutions of the fbsdes in the space l 2 r d r d l 2 r d r k l 2 r d r k d this gives a connection with the weak solutions of quasi linear parabolic pdes finally we study the unique weak solutions of quasi linear elliptic pdes using the solutions of the fbsdes on infinite horizon

backward stochastic partial differential equations with quadratic growth abstract this paper is concerned with the weak solution in analytic sense to the cauchy dirichlet problem of a backward stochastic partial differential equation when the nonhomogeneous term has a quadratic growth in both the gradient of the first unknown and the second unknown existence and uniqueness results are obtained under separate conditions

backward stochastic differential equations associated with the vorticity equations abstract in this paper we derive a non linear version of the feynman kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions we prove the existence global in time and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2 a particular class of terminal values provide via these probabilistic methods solutions for the vorticity equation

forward backward stochastic differential equations and controlled mckean vlasov dynamics the purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the mckean vlasov type motivated by the recent interest in mean field games we highlight the connection and the differences between the two sets of problems we prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control we also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions

general linear quadratic optimal stochastic control problems with random coefficients linear stochastic hamilton systems and backward stochastic riccati equations consider the minimization of the following quadratic cost functional j u e langle mx t x t rangle e int 0 t langle q sx s x s rangle langle n su s u s rangle ds where x is the solution of the following linear stochastic control system eq dx t a tx t b tu t dt sum i 1 d c t ix t d t iu t dw t i cr x 0 h cr u is a square integrable adapted process the problem is conventionally called the stochastic lq the abbreviation of linear quadratic problem we are concerned with the following general case the coefficients a b ci di q n and m are allowed to be adapted processes or random matrices we prove the existence and uniqueness result for the associated riccati equation which in our general case is a backward stochastic differential equation with the generator the drift term being highly nonlinear in the two unknown variables this solves bismut and peng s long standing open problem for the case of a brownian filtration which was initially proposed by the french mathematician j m bismut in 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 we also provide a rigorous derivation of the riccati equation from the stochastic hamilton system this completes the interrelationship between the riccati equation and the stochastic hamilton system as two different but equivalent tools for the stochastic lq problem there are two key points in our arguments the first one is to connect the existence of the solution of the riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system actually we establish their equivalence as a consequence we can construct solutions to a sequence of suitably modified riccati equations in terms of the associated stochastic hamilton systems and the optimal controls the second key point is to establish a new type of a priori estimate for solutions of riccati equations with which we show that the sequence of constructed solutions has a limit which is a solution to the original riccati equation

on the existence of perfect space time codes perfect space time codes are codes for the coherent multiple input multiple output mimo channel they have been called so since they satisfy a large number of design criteria that makes their performances outmatch many other codes in this correspondence we discuss the existence of such codes or more precisely the existence of perfect codes with optimal signal complexity

transmit beamforming in multiple antenna systems with finite rate feedback a vq based approach this paper investigates quantization methods for feeding back the channel information through a low rate feedback channel in the context of multiple input single output miso systems we propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization vq design algorithm the criterion is based on maximizing the mean squared weighted inner product mswip between the optimum and the quantized beamforming vector the performance of systems with quantized beamforming is analyzed for the independent fading case this requires finding the density of the squared inner product between the optimum and the quantized beamforming vector which is obtained by considering a simple approximation of the quantization cell the approximate density function is used to lower bound the capacity loss due to quantization the outage probability and the bit error probability the resulting expressions provide insight into the dependence of the performance of transmit beamforming miso systems on the number of transmit antennas and feedback rate computer simulations support the analytical results and indicate that the lower bounds are quite tight

homogenization of divergence form operators with lower order terms in random media the probabilistic machinery central limit theorem feynman kac formula and girsanov theorem is used to study the homogenization property for pde with second order partial differential operator in divergence form whose coefficients are stationary ergodic random fields furthermore we use the theory of dirichlet forms so that the only conditions required on the coefficients are non degeneracy and boundedness

the dynamic programming equation for second order stochastic target problems motivated by applications in mathematical finance u cetin h m soner and n touzi options hedging for small investors under liquidity costs finance stoch to appear we continue our study of second order backward stochastic equations in this paper we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems in contrast with previous formulations of similar problems we restrict control processes to be continuous this new framework enables us to apply our results to a larger class of models also the resulting derivation is more transparent the main technical tool is the geometric dynamic programming principle in this context and it is proved by using the framework developed in h m soner and n touzi j eur math soc jems 8 2002 pp 201 236

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

application of homogenization and large deviations to a parabolic semilinear equation abstract we study the behavior of the solution of a partial differential equation with a linear parabolic operator with non constant coefficients varying over length scale and nonlinear reaction term of scale 1 the behavior is required as tends to 0 with small compared to we use the theory of backward stochastic differential equations corresponding to the parabolic equation since decreases faster than we may apply the large deviations principle with homogenized coefficients

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric

on weak solutions of forward backward sdes in this paper we continue exploring the notion of weak solution of forward backward stochastic differential equations fbsdes and associated forward backward martingale problems fbmps the main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work ma et al in ann probab 36 6 2092 2125 2008 we consider a general class of non degenerate fbsdes in which all the coefficients are assumed to be essentially only bounded and uniformly continuous and the uniqueness is proved in the space of all the square integrable adapted solutions the standard solution space in the fbsde literature a new notion of semi strong solution is introduced to clarify the relations among different definitions of weak solution in the literature and it is in fact instrumental in our uniqueness proof as a by product we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear pde

mimo precoding with x and y codes we consider a slow fading multiple input multiple output mimo system with channel state information at both the transmitter and receiver a well known precoding scheme is based upon the singular value decomposition svd of the channel matrix which transforms the mimo channel into parallel subchannels despite having low maximum likelihood decoding mld complexity this svd precoding scheme provides a diversity gain which is limited by the diversity gain of the weakest subchannel we therefore propose x and y codes which improve the diversity gain of the svd precoding scheme but maintain the low mld complexity by jointly coding information across a pair of subchannels in particular subchannels with high diversity gain are paired with those having low diversity gain a pair of subchannels is jointly encoded using a 2 2 real matrix which is fixed a priori and does not change with each channel realization for x codes these rotation matrices are parameterized by a single angle while for y codes these matrices are left triangular matrices moreover we propose x y precoders with the same structure as x y codes but with encoding matrices adapted to each channel realization we observed that x codes precoders are good for well conditioned channels while y codes precoders are good for ill conditioned channels

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

constellation precoded multiple beamforming beamforming techniques that employ singular value decomposition svd are commonly used in multi input multi output mimo wireless communication systems in the absence of channel coding when a single symbol is transmitted these systems achieve the full diversity order provided by the channel whereas when multiple symbols are simultaneously transmitted this property is lost when channel coding is employed full diversity order can be achieved for example when bit interleaved coded modulation bicm is combined with this technique full diversity order of nm in an m x n mimo channel transmitting s parallel streams is possible provided a condition on s and the bicm convolutional code rate is satisfied in this paper we present constellation precoded multiple beamforming which can achieve the full diversity order both with bicm coded and uncoded svd systems we provide an analytical proof of this property to reduce the computational complexity of maximum likelihood ml decoding in this system we employ sphere decoding sd we report an sd technique that reduces the computational complexity beyond commonly used approaches to sd this technique achieves several orders of magnitude reduction in computational complexity not only with respect to conventional ml decoding but also with respect to conventional sd

on the complexity of sphere decoding in digital communications sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications in some cases the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure another claim is that although worst case complexity is exponential the expected complexity of the algorithm is polynomial herein we study the expected complexity where the problem size is defined to be the number of symbols jointly detected and our main result is that the expected complexity is exponential for fixed signal to noise ratio snr contrary to previous claims the sphere radius which is a parameter of the algorithm must be chosen to ensure a nonvanishing probability of solving the detection problem this causes the exponential complexity since the squared radius must grow linearly with problem size the rate of linear increase is however dependent on the noise variance and thus the rate of the exponential function is strongly dependent on the snr therefore sphere decoding can be efficient for some snr and problems of moderate size even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size

random time changes and convergence in distribution under the meyer zheng conditions an analog of conditions of meyer and zheng for the relative compactness in the sense of convergence in distribution of a sequence of stochastic processes is formulated for general separable metric spaces and the corresponding notion of convergence is characterized in terms of the convergence in the skorohod topology of time changes of the original processes in addition convergence in distribution under the topology of convergence in measure is discussed and results of jacod memin and metivier on convergence under the skorohod topology are extended

a stable multistep scheme for solving backward stochastic differential equations in this paper we propose a stable multistep scheme on time space grids for solving backward stochastic differential equations in our scheme the integrands which are conditional mathematical expectations derived from the original equations are approximated by using lagrange interpolating polynomials with values of the integrands at multiple time levels they are then numerically evaluated using the gauss hermite quadrature rules and polynomial interpolations on the spatial grids error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions finally various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme

on semi linear degenerate backward stochastic partial differential equations in this paper we study a class of one dimensional degenerate semilinear backward stochastic partial differential equations bspdes for short of parabolic type by establishing some new a priori estimates for both linear and semilinear bspdes we show that the regularity and uniform boundedness of the adapted solution to the semilinear bspde can be determined by those of the coefficients a special feature that one usually does not expect from a stochastic differential equation the proof follows the idea of the so called bootstrap method which enables us to analyze each of the derivatives of the solution under consideration some related results including some comparison theorems of the adapted solutions for semilinear bspdes as well as a nonlinear stochastic feynman kac formula are also given

pricing of american contingent claims with jump stock price and constrained portfolios in this paper we study the problem of pricing american contingent claims in an incomplete market where the stock price process is supposed to be driven by both a wiener process and a poisson random measure and the portfolios are constrained we formulate this problem as to find the minimal solution of a backward stochastic differential equation bsde with constraints we use the penalization method to construct a sequence of bsdes with respect to wiener process and poisson random measure and we show that the solutions of these equations converge to the minimal solution we are interested in finally in the markovian case we characterize the minimal hedging price as the minimal viscosity supersolution of an integral partial differential inequality with constraints

anticipated backward stochastic differential equations driven by the teugels martingales abstract in this paper a class of anticipated backward stochastic differential equations driven by teugels martingales associated with levy process is investigated we obtain the existence and uniqueness of solutions to these equations by means of the fixed point theorem we show that a comparison theorem for this type of absdes also holds under some slight stronger conditions

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

reflected scheme for doubly reflected bsdes with jumps and rcll obstacles we introduce a discrete time reflected scheme to solve doubly reflected backward stochastic differential equations with jumps in short drbsdes driven by a brownian motion and an independent compensated poisson process as in dumitrescu and labart 2014 we approximate the brownian motion and the poisson process by two random walks but contrary to this paper we discretize directly the drbsde without using a penalization step this gives us a fully implementable scheme which only depends on one parameter of approximation the number of time steps n contrary to the scheme proposed in dumitrescu and labart 2014 which also depends on the penalization parameter we prove the convergence of the scheme and give some numerical examples

optimal control problems for stochastic delay evolution equations in banach spaces we consider the optimal control for a banach space valued stochastic delay evolution equation the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of backward stochastic differential equations an application to optimal control of stochastic delay partial differential equations is also given

backward stochastic optimal control with mixed deterministic controller and random controller and its applications in linear quadratic control abstract this paper investigates an optimal control problem driven by backward stochastic differential equation bsde with two controllers one is called deterministic controller and the other one is called random controller necessary and sufficient conditions for the mixed optimal control problem are derived a linear quadratic lq case of the mixed optimal control problem is also studied the mixed optimal controllers are explicitly expressed by the solution of a fully coupled mean field forward backward stochastic differential equation sde one of novel features is that a kind of mean field bsde naturally arises from the research on the mixed optimal control problem finally a product management problem is used to illustrate the theoretical results

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

homogenization of periodic linear degenerate pdes it is well known under the name of periodic homogenization that under a centering condition of the drift a periodic diffusion process on r d converges under diffusive rescaling to a d dimensional brownian motion existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion in this paper we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set as a consequence it is no longer the case that the effective diffusivity matrix is necessarily non degenerate it turns out that provided that some very weak regularity conditions are met the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion in particular this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank we also discuss the application of our results to the homogenization of a class of elliptic and parabolic pdes c 2008 elsevier inc all rights reserved

model reduction with relative multiplicative error bounds and relations to controller reduction balanced model reduction with a priori relative multiplicative error bounds in l sub spl infin norm is studied inverse weighted balanced truncation iwbt is proposed that is a variant of the enns algorithm it is shown that iwbt is equivalent to the balanced stochastic truncation bst this equivalence leads to the extension of an improved relative multiplicative error bound for bst obtained by wang and safonov to general nonsquare transfer matrices natural relations between iwbt and observer based controller reduction are investigated which motivate the simultaneous reduction of the plant and controller robust stability conditions for simultaneous reduction of the plant and the controller are established for the closed loop system consisting of full order plant and reduced order controller as a by product robust stability and stabilization problems are solved for general coprime factor plant descriptions with relative multiplicative h sub spl infin norm bounded uncertainties similar results are obtained for general coprime factors of the feedback controller involving relative multiplicative h sub spl infin norm bounded uncertainties

stochastic pdies with nonlinear neumann boundary conditions and generalized backward doubly stochastic differential equations driven by levy processes in this paper a new class of generalized backward doubly stochastic differential equations gbdsdes in short driven by teugels martingales associated with levy process and the integral with respect to an adapted continuous increasing process is investigated we obtain the existence and uniqueness of solutions to these equations a probabilistic interpretation for solutions to a class of stochastic partial differential integral equations pdies in short with a nonlinear neumann boundary condition is given

perfect space time block codes in this paper we introduce the notion of perfect space time block codes stbcs these codes have full rate full diversity nonvanishing constant minimum determinant for increasing spectral efficiency uniform average transmitted energy per antenna and good shaping we present algebraic constructions of perfect stbcs for 2 3 4 and 6 antennas

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

viability property for a backward stochastic differential equation and applications to partial differential equations in the present paper we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints this property is the so called viability property in a separate section this condition is translated to a class of partial differential equations

the maximum principle for global solutions of stochastic stackelberg differential games national science foundation dms 1303775 research grant council of the hksar cityu 500113

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

bsdes with regime switching weak convergence and applications abstract this paper is concerned with a system of backward stochastic differential equations bsdes with regime switching the bsdes are coupled by a finite state markov chain the underlying markov chain is assumed to have a two time scale or weak and strong interactions structure namely the states of the markov chain can be divided into a number of groups so that the chain jumps rapidly within a group and slowly between the groups it is shown in this paper that the original bsde system can be approximated by a limit system in which the states in each group are aggregated out and replaced by a single state in particular it is proved that the solution of the original bsde system converges weakly under the meyer zheng topology as the fast jump rate goes to infinity the limit process is a solution of aggregated bsdes which can be determined by the corresponding martingale problem the results are applied to a set of partial differential equations and used to validate their convergence to the corresponding limit system finally a numerical example is given to demonstrate the approximation results

parametric h sub spl infin loopshaping and weighted mixed sensitivity minimization generalized h sub spl infin loopshaping is developed for a particular weighted mixed sensitivity minimization task a free parameter is introduced to tradeoff minimizations of sensitivity and complementary sensitivity in the h sub spl infin performance index a quadratically convergent algorithm is derived for computing the optimal performance index and a state space solution is obtained for suboptimal controllers these results are applicable to stabilization of uncertain systems where normalized coprime factors involve parametrically weighted h sub spl infin norm bounded perturbations controller reduction is also investigated for the parametric h sub spl infin loopshaping robust stability condition for using the reduced order controller is established and an estimate for the associated h sub spl infin performance is obtained similar results are derived for normalized linear quadratic gaussian lqg control

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

mean variance hedging under partial information we consider the mean variance hedging problem under partial information the underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available we show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term which is formulated in terms of observable processes we prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation

numerical methods for a class of nonlocal diffusion problems with the use of backward sdes we propose a novel numerical approach for nonlocal diffusion equations du et al 2012 with integrable kernels based on the relationship between the backward kolmogorov equation and backward stochastic differential equations bsdes driven by levy jumps processes the nonlocal diffusion problem under consideration is converted to a bsde for which numerical schemes are developed as a stochastic approach the proposed method completely avoids the challenge of iteratively solving non sparse linear systems arising from the nature of nonlocality this allows for embarrassingly parallel implementation and also enables adaptive approximation techniques to be incorporated in a straightforward fashion moreover our method recovers the convergence rates of classic deterministic approaches e g finite element or collocation methods due to the use of high order temporal and spatial discretization schemes in addition our approach can handle a broad class of problems with general inhomogeneous forcing terms as long as they are globally lipschitz continuous rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach

sharp derivative bounds for solutions of degenerate semi linear partial differential equations abstract the paper is a continuation of the kusuoka stroock programme of establishing smoothness properties of solutions of possibly degenerate partial differential equations by using probabilistic methods we analyze here a class of semi linear parabolic partial differential equations for which the linear part is a second order differential operator of the form v 0 i 1 n v i 2 where v 0 v n are first order differential operators that satisfy the so called ufg condition see kusuoka and stroock 1987 16 which is weaker than the hormander one specifically we prove that the bounds of the higher order derivatives of the solution along the vector fields coincide with those obtained in the linear case when the boundary condition is lipschitz continuous but that the asymptotic behavior of the derivatives may change because of the simultaneity of the nonlinearity and of the degeneracy when the boundary condition is of polynomial growth and measurable only

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

the maximum principles for stochastic recursive optimal control problems under partial information a maximum principle for partially observed stochastic recursive optimal control problems is obtained under the assumption that control domains are not necessarily convex and forward diffusion coefficients do not contain control variables such kind of recursive optimal control problems have wide applications in finance and economics such as recursive utility optimization and principal agent problems by virtue of a classical spike variational approach and a filtering technique the maximum principle is obtained and the related adjoint processes are characterized by the solutions of forward backward stochastic differential equations in finite dimensional spaces then our theoretical results are applied to study a partially observed linear quadratic recursive optimal control problem in addition for the case with initial and terminal state constraints the corresponding maximum principle is also obtained by using ekeland s variational principle

a fourier cosine method for an efficient computation of solutions to bsdes we develop a fourier method to solve backward stochastic differential equations bsdes a general theta discretization of the time integrands leads to an induction scheme with conditional expectations these are approximated by using fourier cosine series expansions relying on the availability of a characteristic function the method is applied to bsdes with jumps numerical experiments demonstrate the applicability of bsdes in financial and economic problems and show fast convergence of our efficient probabilistic numerical method

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

achieving the welch bound with difference sets consider a codebook containing n unit norm complex vectors in a k dimensional space in a number of applications the codebook that minimizes the maximal cross correlation amplitude i sub max is often desirable relying on tools from combinatorial number theory we construct analytically optimal codebooks meeting in certain cases the welch lower bound when analytical constructions are not available we develop an efficient numerical search method based on a generalized lloyd algorithm which leads to considerable improvement on the achieved i sub max over existing alternatives we also derive a composite lower bound on the minimum achievable i sub max that is effective for any codebook size n

multiple beamforming with perfect coding perfect space time block codes pstbcs achieve full diversity full rate nonvanishing constant minimum determinant uniform average transmitted energy per antenna and good shaping however the high decoding complexity is a critical issue for practice when the channel state information csi is available at both the transmitter and the receiver singular value decomposition svd is commonly applied for a multiple input multiple output mimo system to enhance the throughput or the performance in this paper two novel techniques perfect coded multiple beamforming pcmb and bit interleaved coded multiple beamforming with perfect coding bicmb pc are proposed employing both pstbcs and svd with and without channel coding respectively with csi at the transmitter csit the decoding complexity of pcmb is substantially reduced compared to a mimo system employing pstbc providing a new prospect of csit especially because of the special property of the generation matrices pcmb provides much lower decoding complexity than the state of the art svd based uncoded technique in dimensions 2 and 4 similarly the decoding complexity of bicmb pc is much lower than the state of the art svd based coded technique in these two dimensions and the complexity gain is greater than the uncoded case moreover these aforementioned complexity reductions are achieved with only negligible or modest loss in performance

necessary and sufficient conditions for near optimality in stochastic control of fbsdes abstract we consider control problems for systems governed by a nonlinear forward backward stochastic differential equation fbsde we establish necessary as well as sufficient conditions for near optimality satisfied by all near optimal controls these conditions are described by two adjoint processes corresponding to the forward and backward components and a nearly maximum condition on the hamiltonian the proof of the main result is based on ekeland s variational principle and some estimates on the state and the adjoint processes with respect to the control variable as is well known optimal controls may fail to exist even in simple cases this justifies the use of near optimal controls which exist under minimal assumptions and are sufficient in most practical cases moreover since there are many nearly optimal controls it is possible to choose suitable ones that are convenient for implementation

systematic design of unitary space time constellations we propose a systematic method for creating constellations of unitary space time signals for multiple antenna communication links unitary space time signals which are orthonormal in time across the antennas have been shown to be well tailored to a rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients the signals can achieve low probability of error by exploiting multiple antenna diversity because the fading coefficients are not known the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum euclidean distance norm our construction begins with the first signal in the constellation an oblong complex valued matrix whose columns are orthonormal and systematically produces the remaining signals by successively rotating this signal in a high dimensional complex space this construction easily produces large constellations of high dimensional signals we demonstrate its efficacy through examples involving one two and three transmitter antennas

dirichlet forms and semilinear elliptic equations with measure data we propose a probabilistic definition of solutions of semilinear elliptic equations with possibly nonlocal operators associated with regular dirichlet forms and with measure data using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right hand side of the equation is monotone and satisfies mild integrability assumption and the measure is smooth we also study regularity of solutions under the assumption that the measure is smooth and has finite total variation some applications of our general results are given

spatio temporal coding for wireless communication multipath signal propagation has long been viewed as an impairment to reliable communication in wireless channels this paper shows that the presence of multipath greatly improves achievable data rate if the appropriate communication structure is employed a compact model is developed for the multiple input multiple output mimo dispersive spatially selective wireless communication channel the multivariate information capacity is analyzed for high signal to noise ratio snr conditions the mimo channel can exhibit a capacity slope in bits per decibel of power increase that is proportional to the minimum of the number multipath components the number of input antennas or the number of output antennas this desirable result is contrasted with the lower capacity slope of the well studied case with multiple antennas at only one side of the radio link a spatio temporal vector coding stvc communication structure is suggested as a means for achieving mimo channel capacity the complexity of stvc motivates a more practical reduced complexity discrete matrix multitone dmmt space frequency coding approach both of these structures are shown to be asymptotically optimum an adaptive lattice trellis coding technique is suggested as a method for coding across the space and frequency dimensions that exist in the dmmt channel experimental examples that support the theoretical results are presented

singular homogenization with stationary in time and periodic in space coefficients we study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential in a general stationary and mixing random media which is periodic in space it is shown that a solution of the corresponding cauchy problem converges in law to a solution of a limit stochastic pde 2005 published by elsevier inc msc 35k57 60h15

a pontryagin s maximum principle for non zero sum differential games of bsdes with applications this technical note is concerned with a maximum principle for a new class of non zero sum stochastic differential games the most distinguishing feature compared with the existing literature is that the game systems are described by backward stochastic differential equations bsdes this kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities we establish a necessary condition and a sufficient condition in the form of maximum principle for open loop equilibrium point of the foregoing games respectively to explain the theoretical results we use them to study a financial 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

precoding by pairing subchannels to increase mimo capacity with discrete input alphabets we consider gaussian multiple input multiple output mimo channels with discrete input alphabets we propose a non diagonal precoder based on the x codes in to increase the mutual information the mimo channel is transformed into a set of parallel subchannels using singular value decomposition svd and x codes are then used to pair the subchannels x codes are fully characterized by the pairings and a 2 2 real rotation matrix for each pair parameterized with a single angle this precoding structure enables us to express the total mutual information as a sum of the mutual information of all the pairs the problem of finding the optimal precoder with the above structure which maximizes the total mutual information is solved by i optimizing the rotation angle and the power allocation within each pair and ii finding the optimal pairing and power allocation among the pairs it is shown that the mutual information achieved with the proposed pairing scheme is very close to that achieved with the optimal pre coder by cruz et al and is significantly better than mercury waterfllling strategy by lozano et al our approach greatly simplifies both the precoder optimization and the detection complexity making it suitable for practical applications

bsdes with polynomial growth generators in a defaultable market abstract in this paper we study a backward stochastic differential equation bsde for short driven by a brownian motion and a jump martingale in a defaultable setting with a polynomial growth generator this kind of bsde has important applications to the defaultable market to demonstrate this in a strict way we first prove the existence uniqueness and uniform p p 2 estimate of its solution then two examples are given to illustrate its applications to recursive utility and contingent claim hedging

singular control and optimal stopping of spdes and backward spdes with reflection we consider general singular control problems for random fields given by a stochastic partial differential equation spde we show that under some conditions the optimal singular control can be identified with the solution of a coupled system of spde and a reflected backward spde rbspde as an illustration we apply the result to a singular optimal harvesting problem from a population whose density is modeled as a stochastic reaction diffusion equation existence and uniqueness of solutions of rbspdes are established as well as comparison theorems we then establish a relation between rbspdes and optimal stopping of spdes and apply the result to a risk minimizing stopping problem

multiscale problems and homogenization for second order hamilton jacobi equations we prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic pdes as a special case we cover the iterated homogenization for such equations with oscillating initial data explicit examples among others are the two scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n scale homogenization of uniformly parabolic fully nonlinear pdes

design and analysis of mimo spatial multiplexing systems with quantized feedback this paper investigates the problem of transmit beamforming in multiple antenna spatial multiplexing sm systems employing a finite rate feedback channel assuming a fixed number of spatial channels and equal power allocation we propose a new criterion for designing the codebook of beamforming matrices that is based on minimizing an approximation to the capacity loss resulting from the limited rate in the feedback channel using the criterion we develop an iterative design algorithm that converges to a locally optimum codebook under the independent identically distributed channel and high signal to noise ratio snr assumption the effect on channel capacity of the finite bit representation of the beamforming matrix is analyzed central to this analysis is the complex multivariate beta distribution and tractable approximations to the voronoi regions associated with the code points furthermore to compensate for the degradation due to the equal power allocation assumption we propose a multimode sm transmission strategy wherein the number of data streams is determined based on the average snr this approach is shown to allow for effective utilization of the feedback bits resulting in a practical and efficient multiple input multiple output system design

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results