the stochastic maximum principle in singular optimal control with recursive utilities abstract in this paper we consider stochastic recursive optimal control problem in which the control variable has two components with the first absolutely continuous and the second singular the control domain of the first component needs not to be convex by using a spike variation on the absolutely continuous part of the control and a convex perturbation on the singular one respectively we obtain a stochastic maximum principle of the optimal control also we give the relationship of the backward variational equation the adjoint equation and forward variational equation

first and second order necessary conditions for stochastic optimal controls abstract the main purpose of this paper is to establish the first and second order necessary optimality conditions for stochastic optimal controls using the classical variational analysis approach the control system is governed by a stochastic differential equation in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex only one adjoint equation is introduced to derive the first order necessary condition while only two adjoint equations are needed to state the second order necessary conditions for stochastic optimal controls

pointwise second order necessary conditions for stochastic optimal controls part i the case of convex control constraint this paper is the first part of our series of work to establish pointwise second order necessary conditions for stochastic optimal controls in this part both drift and diffusion terms may contain the control variable but the control region is assumed to be convex under some assumptions in terms of the malliavin calculus we establish the desired necessary conditions for stochastic singular optimal controls in the classical sense

discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

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

brief paper stochastic maximum principle in the mean field controls in buckdahn djehiche li and peng 2009 the authors obtained mean field backward stochastic differential equations bsdes in a natural way as a limit of some highly dimensional system of forward and backward sdes corresponding to a great number of particles or agents the objective of the present paper is to deepen the investigation of such mean field bsdes by studying their stochastic maximum principle this paper studies the stochastic maximum principle smp for mean field controls which is different from the classical ones this paper deduces an smp in integral form and it also gets under additional assumptions necessary conditions as well as sufficient conditions for the optimality of a control as an application this paper studies a linear quadratic stochastic control problem of mean field type

hochschild cohomology of abelian categories and ringed spaces this paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors we show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of hochschild cohomology for abelian categories we then show that this hochschild cohomology coincides with the one defined by gerstenhaber schack and swan in the case of module categories over diagrams and schemes and also with the hochschild cohomology for exact categories introduced recently by keller in addition we show in complete generality that hochschild cohomology satisfies a mayer vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf

discontinuous galerkin error estimation for hyperbolic problems on unstructured triangular meshes we extend the error analysis of adjerid and baccouch 1 2 for the discontinuous galerkin discretization error to variable coefficient linear hyperbolic problems as well as nonlinear hyperbolic problems on unstructured meshes we further extend this analysis to transient hyperbolic problems and prove that the local superconvergence results presented in 1 still hold for both steady and transient variable coefficient linear and nonlinear problems this local error analysis allows us to construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on each element of general unstructured meshes we illustrate the superconvergence and the efficiency of our a posteriori error estimates by showing computational results for several linear and nonlinear numerical examples

robust controller design for stochastic nonlinear systems via convex optimization this paper presents a new nonlinear control design framework called convex optimization based stochastic steady state tracking error minimization cvstem for a class of ito stochastic nonlinear systems including stochastic lagrangian systems this state feedback controller design involves the use of multiple state dependent coefficient sdc models of a nonlinear system equation and computes its controller parameters by solving a convex optimization problem in order to minimize an upper bound of the steady state tracking error this problem is shown to be equivalent to the original nonlinear optimization problem of minimizing the upper bound the exponential stability and robustness properties due to the state dependent riccati inequality are proven using stochastic contraction analysis discrete time stochastic incremental contraction analysis with respect to a state and time dependent metric is also presented along with its explicit connection to continuous time contraction analysis a result of simulation is used to validate the superiority of the proposed controller compared to a known nonlinear controller a pid controller and an h infinity controller

maximum principle for mean field jump diffusion stochastic delay differential equations and its application to finance abstract this paper investigates a stochastic optimal control problem with delay and of mean field type where the controlled state process is governed by a mean field jump diffusion stochastic delay differential equation two sufficient maximum principles and one necessary maximum principle are established for the underlying system as an application a bicriteria mean variance portfolio selection problem with delay is studied to demonstrate the effectiveness and potential of the proposed techniques under certain conditions explicit expressions are provided for the efficient portfolio and the efficient frontier which are as elegant as those in the classical mean variance problem without delays

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

maximum principle for optimal control problems of forward backward regime switching system and applications abstract in this paper we derive the stochastic maximum principle for optimal control problems of the forward backward markovian regime switching system the control system is described by forward backward sdes and modulated by continuous time finite state markov chains we first obtain the necessary and sufficient conditions for the optimal control thereafter we apply the maximum principle to recursive utility investment consumption problems and lq problems with markovian regime switching

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

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

a stochastic maximum principle for backward control systems with random default time this paper establishes a necessary and sufficient stochastic maximum principle for backward systems where the state processes are governed by jump diffusion backward stochastic differential equations with random default time an application of the sufficient stochastic maximum principle to an optimal investment and capital injection problem in the presence of default risk is discussed

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

on backward stochastic evolution equations in hilbert spaces and optimal control abstract in this paper a new result on the existence and uniqueness of the adapted solution to a backward stochastic evolution equation in hilbert spaces under a non lipschitz condition is established the applicability of this result is then illustrated in a discussion of a concrete backward stochastic partial differential equation furthermore a stochastic maximum principle for optimal control problems of stochastic systems governed by backward stochastic evolution equations in hilbert spaces is obtained

a posteriori error analysis of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present asymptotically exact a posteriori error estimators of the local discontinuous galerkin ldg method for the nonlinear sine gordon equation in one space dimension we apply our recent optimal l 2 error estimates and superconvergence results baccouch 2018 to show that the ldg errors on each element can be split into two parts the first part is proportional to the p 1 degree radau polynomial and the second part converges with order p 3 2 in the l 2 norm when piecewise polynomials of degree at most p are used this new result is used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element the proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element we further prove that for smooth solutions these a posteriori error estimates for the solution and its derivative at a fixed time converge to the exact spatial errors in the l 2 norm under mesh refinement the theoretical order of convergence is proved to be p 3 2 while our computational results show higher o h p 2 convergence rate finally we prove that the global effectivity index converges to unity at o h 1 2 rate several numerical results are presented to validate the theoretical results

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

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

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

a posteriori error analysis for numerical approximations of friedrichs systems the global error of numerical approximations for symmetric positive systems in the sense of friedrichs is decomposed into a locally created part and a propagating component residual based two sided local a posteriori error bounds are derived for the locally created part of the global error these suggest taking the l 2 norm as well as weaker dual norms of the computable residual as local error indicators the dual graph norm of the residual vec r h is further bounded from above and below in terms of the l 2 norm of h vec r h where h is the local mesh size the theoretical results are illustrated by a series of numerical experiments

second order taylor expansion for backward doubly stochastic control system we consider the second order taylor expansion for backward doubly stochastic control system the results are obtained under no restriction on the convexity of control domain moreover the control variable is allowed in the drift coefficient and the diffusion coefficient

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

martingale approach to stochastic differential games of control and stopping we develop a martingale approach for studying continuous time stochastic differential games of control and stopping in a non markovian framework and with the control affecting only the drift term of the state process under appropriate conditions we show that the game has a value and construct a saddle pair of optimal control and stopping strategies crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities

applicable stochastic control from theory to practice stochastic control has come of age it is currently applied in a wide variety of fields spanning cancer research and chemotherapy economics and finance engineering management and many other areas of scientific and social investigation both theoretical models formulated to elucidate some characteristics of a problem under study and practical applications which have sought to provide specific answers to well stated problems have been the subjects of intense research for example economists have simultaneously groped with the underlying theoretical constructs justifying the adoption of particular econometric models while at the same time attempting to devise algorithms and optimization approaches to provide economic policy guidelines based on these models in the same spirit t remendous interest in options management which are based on stochastic dynamic models has stimulated much research and numerical analyses not only in universities but also in the back rooms of trading firms the usefulness of stochastic control was recognized once it became clear that it can be used as a unifying framework for treating multi stage or multi period optimization problems subject to stochastic disturbances however just as it is convenient as a formalism to formulate such problems it is becoming increasingly clear that it is practically difficult to solve them there are in fact only few problems one can solve completely and obtain optimal solutions for this reason there is today an increased awareness that a broad range of techniques including stochastic approximations perturbation techniques markov decision processes etc are an essential part of stochastic control kits it is in such a spirit that computer software such as pandore of inria the adaptive control algorithms for econometric models developed by kendrick and many others has been devised providing thereby computer aided expertise for managing stochastic control problems this software is essential even after the great effort applied to solving such problems over the last three decades ever since bellman s principle of optimality there are very few non trivial problems we can solve theoretically these include the well known linear quadratic stochastic control scp problem some version of the multi armed bandit problem and some specific linear optimization problems for example even the adaptive control problems studied over the last few years by kendrick and his co workers and presumed to be convex are believed today to include some unspecified nonconvexities private communication by d kendrick and hans amman in this spirit the intent in this paper is to introduce salient aspects of stochastic control which are useful for applications in management a review of applications can be found in neck 1984 for example as well as in books such as tapiero 1977 1988 bensoussan hurst and naslund 1974 bensoussan kleindorfer and tapiero 1978 1980 sethi and thompson 1981 kamien and schwarz 1981 malliaris and brock 1982 and whittle 1982 further due to an awesome amount of research published in this area appearing under various connotations such a task has become unwieldy rather

moduli of non commutative polarized schemes we construct using geometric invariant theory a quasi projective deligne mumford stack of stable graded algebras we also construct a derived enhancement which classifies twisted bundles of stable graded a infty algebras the tangent complex of the derived scheme is given by graded hochschild cohomology which we relate to ordinary hochschild cohomology we obtain a version of hilbert stability for non commutative projective schemes

stochastic optimal control for backward stochastic partial differential systems abstract this paper studies optimal controls for a class of backward stochastic partial differential systems in the abstract evolution form under the assumption of a convex control domain necessary and sufficient conditions for an admissible control to be optimal are derived in the form of stochastic maximum principles by means of a convex variation method and a duality technique as an application the optimal control for a linear backward stochastic evolution equation bsee with quadratic cost criteria called bseelq problem is discussed and the corresponding optimal control is characterized via the stochastic hamilton system which is a linear full coupled forward backward stochastic evolution equation fbsee and consists of the state equation the adjoint equation and the dual presentation of the optimal control

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

a posteriori error estimates for a discontinuous galerkin method applied to elliptic problems log number r74 abstract a posteriori error estimates for locally mass conservative methods for subsurface flow are presented these methods are based on discontinuous approximation spaces and referred to as discontinuous galerkin methods in the case where penalty terms are added to the bilinear form one obtains the nonsymmetric interior penalty galerkin methods in a previous work we proved optimal rates of convergence of the methods applied to elliptic problems here h adaptivity is investigated for flow problems in 2d we derive global explicit estimators of the error in the l 2 norm and we numerically investigate an implicit indicator of the error in the energy norm model problems with discontinuous coefficients are considered

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

stochastic controls with terminal contingent conditions abstract this paper considers a nonlinear stochastic control problem where the system dynamics is a controlled nonlinear backward stochastic differential equation and the state must coincide with a given random vector at the terminal time a necessary condition of optimality in the form of a global maximum principle as well as a sufficient condition of optimality are presented the general result is also applied to backward linear quadratic control problem and an optimal control is obtained explicitly as a feedback of the solution to a forward backward equation finally a nonlinear problem with additional integral constraints is discussed and it is shown that the duality gap is zero under the slater condition

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

superconvergence and a posteriori error estimates for the ldg method for convection diffusion problems in one space dimension in this paper we investigate the superconvergence properties of the local discontinuous galerkin ldg method applied to transient convection diffusion problems in one space dimension we show that the leading terms of the local discretization errors for the p degree ldg solution and its spatial derivative are proportional to p 1 degree right and left radau polynomials respectively thus the discretization errors for the p degree ldg solution and its spatial derivative are o h p 2 superconvergent at the roots of p 1 degree right and left radau polynomials respectively the superconvergence results are used to construct asymptotically correct a posteriori error estimates these error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element numerical computations suggest that these a posteriori ldg error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at o h p 3 and o h p 2 rates respectively we also show that the global effectivity indices for the solution and its derivative in the l 2 norm converge to unity at o h 2 and o h rates respectively finally we show that the ldg method combined with the a posteriori error estimation procedure yields both accurate error estimates and o h p 2 superconvergent solutions our proofs are valid for arbitrary regular meshes and for p p polynomials with p 1 and for periodic dirichlet and mixed dirichlet neumann boundary conditions several numerical simulations are performed to validate the theory

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

a superconvergent local discontinuous galerkin method for the second order wave equation on cartesian grids abstract in this paper we propose and analyze a new superconvergent local discontinuous galerkin ldg method equipped with an element residual error estimator for the spatial discretization of the second order wave equation on cartesian grids we prove the l 2 stability the energy conserving property and optimal l 2 error estimates for the semi discrete formulation in particular we identify special numerical fluxes for which the l 2 norm of the solution and its gradient are of order p 1 when tensor product polynomials of degree at most p are used we further perform a local error analysis and show that the leading term of the ldg error is spanned by two p 1 degree right radau polynomials in the x and y directions thus the ldg solution is o h p 2 superconvergent at radau points obtained as a tensor product of the roots of p 1 degree right radau polynomial furthermore numerical computations show that the first component of the solution s gradient is o h p 2 superconvergent at tensor product of the roots of left radau polynomial in x and right radau polynomial in y while the second component is o h p 2 superconvergent at the tensor product of the roots of the right radau polynomial in x and left radau polynomial in y computational results indicate that global superconvergence holds for ldg solutions we use the superconvergence results to construct a posteriori ldg error estimates these error estimates are computationally simple and are obtained by solving local steady problems with no boundary condition on each element finally we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement

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

sufficient conditions of optimality for stochastic systems with controllable diffusions this paper studies optimal controls for systems governed by ito s stochastic differential equations both the drift and diffusion terms of the equations are allowed to depend on controls and the systems are allowed to be degenerate it is shown that the necessary conditions of optimality namely the maximum conditions in terms of the spl hscr function which is a generalization of the usual hamiltonian and is quadratic with respect to the diffusion coefficients along with some convexity conditions constitute sufficient conditions of optimality for such controlled systems

forward backward linear quadratic stochastic optimal control problem with delay abstract this paper is concerned with one kind of forward backward linear quadratic stochastic control problem whose system is described by a linear anticipated forward backward stochastic differential delayed equation the explicit form of the optimal control is derived optimal state feedback regulators are studied in two special cases for the case with delay in just the control variable the optimal state feedback regulator is obtained by the riccati equation for the other case with delay in just the state variable the optimal state feedback regulator is analyzed by the value function approach

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

maximum principle for forward backward sdes with a general cost functional abstractin this study we consider a stochastic optimal control problem whose state variables are described by the system of forward and backward stochastic differential equations with a general cost functional which relies on the global terminal condition in the framework of frechet derivatives we derive the corresponding maximum principle via constructing a series of adjoint equations which need to be solved step by step

backward stochastic differential equations with constraints on the gains process we consider backward stochastic differential equations with convex constraints on the gains or intensity of noise process existence and uniqueness of a minimal solution are established in the case of a drift coefficient which is lipschitz continuous in the state and gains processes and convex in the gains process it is also shown that the minimal solution can be characterized as the unique solution of a functional stochastic control type equation this representation is related to the penalization method for constructing solutions of stochastic differential equations involves change of measure techniques and employs notions and results from convex analysis such as the support function of the convex set of constraints and its various properties

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

superconvergence of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present superconvergence results for the local discontinuous galerkin ldg method for the sine gordon nonlinear hyperbolic equation in one space dimension we identify a special numerical flux and a suitable projection of the initial conditions for the ldg scheme for which the l 2 norm of the ldg solution and its spatial derivative are of order p 1 when piecewise polynomials of degree at most p are used our numerical experiments demonstrate optimal order of convergence we further prove superconvergence toward particular projections of the exact solutions more precisely we prove that the ldg solution and its spatial derivative are o h p 3 2 super close to particular projections of the exact solutions while computational results show higher o h p 2 convergence rate our analysis is valid for arbitrary regular meshes and for p p polynomials with arbitrary p 1 numerical experiments validating these theoretical results are presented

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

existence of optimal controls for systems driven by fbsdes abstract we prove the existence of optimal relaxed controls as well as strict optimal controls for systems governed by non linear forward backward stochastic differential equations fbsdes our approach is based on weak convergence techniques for the associated fbsdes in the jakubowski s topology and a suitable skorokhod representation theorem

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

optimal replication of random vectors by ordinary integrals abstract we consider a problem of replication of random vectors by ordinary integrals in the setting when an underlying random variable is generated by a wiener process the goal is to find an optimal adapted process such that its cumulative integral at a fixed terminal time matches this variable the optimal process has to be minimal in an integral norm

a new risk sensitive maximum principle in this paper a new maximum principle for the risk sensitive control problem is established one important feature of this result is that it applies to systems in which the diffusion term may depend on the control such control dependence gives rise to interesting phenomena not observed in the usual setting where control independence of the diffusion term is assumed in particular there is an additional second order adjoint equation and additional terms in the maximum condition that involve this second order process as well as the risk sensitive parameter moreover contrary to a conventional maximum principle the first order adjoint equation involved in our maximum principle is a nonlinear equation an advantage of considering this new type of adjoint equation is that the risk sensitive maximum principle derived is similar in form to its risk neutral counterpart the approach is based on the logarithmic transformation and the relationship between the adjoint variables and the value function as an example a linear quadratic risk sensitive problem is solved using the maximum principle derived

time inconsistent stochastic linear quadratic control characterization and uniqueness of equilibrium in this paper we continue our study on a general time inconsistent stochastic linear quadratic control problem originally formulated in y hu h jin and x y zhou siam j control optim 50 2012 pp 1548 1572 we derive a necessary and 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 prove that the explicit equilibrium control constructed in y hu h jin and x y zhou siam j control optim 50 2012 pp 1548 1572 is indeed unique our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the lebesgue differentiation theorem finally we show that the equilibrium strategy is unique for 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 pro

optimal control problem of backward stochastic differential delay equation under partial information abstract in this paper we study the optimal control problem of backward stochastic differential delay equation under partial information a class of time advanced stochastic differential equations asdes is introduced as the adjoint process via duality relation by means of asdes we suggest a maximum principle of this problem and obtain necessary and sufficient conditions of optimality we apply the theoretical results to study linear quadratic optimal control problem and obtain a forward backward stochastic differential filtering equation fbsdfe with time advanced forward equation that leads to optimal control at the same time some auxiliary filtering results and properties about time advanced sdes are also presented

the relaxed optimal control problem for mean field sdes systems and application the present study deals with a new approach of optimal control problems where the state equation is a mean field stochastic differential equation and the set of strict classical controls need not be convex and the diffusion coefficient depends on the term control our consideration is based on only one adjoint process and the necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle are obtained with application to linear quadratic stochastic control problem with mean field type

optimality variational principle for controlled forward backward stochastic differential equations with mixed initial terminal conditions an optimal control problem for general coupled forward backward stochastic differential equations fbsdes with mixed initial terminal conditions is considered the control domain is not assumed to be convex and the control appears in the diffusion coefficient of the forward equation necessary conditions of pontraygin s type for the optimal controls are derived by means of spike variation techniques

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

a local discontinuous galerkin method for the second order wave equation abstract in this paper we present new superconvergence results for the local discontinuous galerkin ldg method applied to the second order scalar wave equation in one space dimension numerical experiments show o h p 1 l 2 convergence rate for the ldg solution and o hp 2 superconvergent solutions at radau points more precisely a local error analysis reveals that at a fixed time t the leading terms of the discretization errors for the solution and its derivative using p degree polynomial approximations are proportional to the p 1 degree right radau and p 1 degree left radau polynomials respectively thus the p degree ldg solution is o hp 2 superconvergent at the roots of the p 1 degree right radau polynomial and the derivative of the ldg solution is o hp 2 superconvergent at the roots of the p 1 degree left radau polynomial these results are used to construct simple efficient and asymptotically correct a posteriori error estimates in regions where solutions are smooth finally we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement

a superconvergence result for discontinuous galerkin methods applied to elliptic problems this paper presents a theoretical and numerical study of a class of discontinuous galerkin methods that shows the approximation of the gradient superconverges at the zeros of the legendre polynomials on a model 1d elliptic problem numerical experiments validate the theoretical results

a maximum principle for optimal control of stochastic evolution equations a general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations the control variable and linear unbounded operators act in both drift and diffusion terms and the control set need not be convex

well posedness and regularity of backward stochastic volterra integral equations backward stochastic volterra integral equations bsvies for short are studied notion of adapted m solution is introduced well posedness of bsvies is established and some regularity results are proved for the adapted m solutions via malliavin calculus a pontryagin type maximum principle is presented for optimal controls of stochastic volterra integral equations

a maximum principle for fully coupled stochastic control systems of mean field type abstract the present paper considers an optimal control problem for fully coupled forward backward stochastic differential equations fbsdes of mean field type in the case of controlled diffusion coefficient moreover the control domain is not assumed to be convex by virtue of a reduction method we establish the necessary optimality conditions of pontryagin s type as an application a linear quadratic stochastic control problem is studied

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

a maximum principle for partial information backward stochastic control problems with applications this paper studies the partial information control problems of backward stochastic systems there are three major contributions made in this paper i first we obtain a new stochastic maximum principle for partial information control problems our method relies on a direct calculation of the derivative of the cost functional ii second we introduce two classes of partial information linear quadratic backward control problems for the first time and then investigate them using the maximum principle complete and explicit solutions are obtained in terms of some forward and backward stochastic differential filtering equations iii last but not least we study a class of full information stochastic pension fund optimization problems which can be viewed as a special case of our general partial information ones applying the aforementioned maximum principle we derive the optimal contribution policy in closed form and present some related economic remarks

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

maximum principle for singular stochastic control problems in this paper an optimal singular stochastic control problem is considered for this model a general stochastic maximum principle is obtained by using a time transformation this is the first version of the stochastic maximum principle that covers nonlinear cases

optimal energy conserving local discontinuous galerkin methods for second order wave equation in heterogeneous media abstract solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering the design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration in this paper we focus on multi dimensional wave problems and consider linear second order wave equation in heterogeneous media we develop and analyze an ldg method in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy compatible high order energy conserving time integrators are also proposed the optimal error estimates and the energy conserving property are proved for the semi discrete methods our numerical experiments demonstrate optimal rates of convergence and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property

superconvergence of the numerical traces of discontinuous galerkin and hybridized methods for convection diffusion problems in one space dimension in this paper we uncover and study a new superconvergence property of a large class of finite element methods for one dimensional convection diffusion problems this class includes discontinuous galerkin methods defined in terms of numerical traces discontinuous petrov galerkin methods and hybridized mixed methods we prove that the so called numerical traces of both variables superconverge at all the nodes of the mesh provided that the traces are conservative that is provided they are single valued in particular for is local discontinuous galerkin method we show that the superconvergence is order 2p 1 when polynomials of degree at most p are used extensive numerical results verifying our theoretical results are displayed

a nonzero sum differential game of bsde with time delayed generator and applications in this technical note a nonzero sum differential game is studied where the state dynamics follows a backward stochastic differential equation with time delayed generator an arrow s sufficient condition for open loop equilibrium point is proved a linear quadratic differential game and a pension fund management problem with time delayed surplus are discussed as the applications of the sufficient optimality condition