kalman bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems abstract this paper is concerned with kalman bucy filtering problems of a forward and backward stochastic system which is a hamiltonian system arising from a stochastic optimal control problem there are two main contributions worthy pointing out one is that we obtain the kalman bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation the other is that we develop a backward separation technique which is different to wonham s separation theorem to study a partially observed recursive optimal control problem this new technique can also cover some more general situation such as a partially observed linear quadratic non zero sum differential game problem is solved by it we also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases

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

a form of schwarz s lemma and a bound for the kobayashi metric on convex domains abstract we present a form of schwarz s lemma for holomorphic maps between convex domains d 1 and d 2 this result provides a lower bound on the distance between the images of relatively compact subsets of d 1 and the boundary of d 2 this is a natural improvement of an old estimate by bernal gonzalez that takes into account the geometry of d 1 using similar techniques we also provide a new estimate for the kobayashi metric on bounded convex domains

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

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

gromov hyperbolicity and the kobayashi metric on convex domains of finite type in this paper we prove necessary and sufficient conditions for the kobayashi metric on a convex domain to be gromov hyperbolic in particular we show that for convex domains with c infty boundary being of finite type in the sense of d angelo is equivalent to the gromov hyperbolicity of the kobayashi metric we also show that bounded domains which are locally convexifiable and have finite type in the sense of d angelo have gromov hyperbolic kobayashi metric the proofs use ideas from the theory of the hilbert metric

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

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

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

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

goldilocks domains a weak notion of visibility and applications in this paper we introduce a new class of domains in complex euclidean space called goldilocks domains and study their complex geometry these domains are defined in terms of a lower bound on how fast the kobayashi metric grows and an upper bound on how fast the kobayashi distance grows as one approaches the boundary strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this goldilocks condition but we also present families of goldilocks domains that have low boundary regularity or have boundary points of infinite type we will show that the kobayashi metric on these domains behaves in some sense like a negatively curved riemannian metric in particular it satisfies a visibility condition in the sense of eberlein and o neill this behavior allows us to prove a variety of results concerning boundary extension of maps and to establish wolff denjoy theorems for a wide collection of domains c 2017 elsevier inc all rights reserved

solvability of indefinite stochastic riccati equations and linear quadratic optimal control problems abstract a new approach to study the indefinite stochastic linear quadratic lq optimal control problems which we called the equivalent cost functional method is introduced by yu 2013 in the setup of hamiltonian system on the other hand another important issue along this research direction is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic riccati equations as the response this paper continues to develop the equivalent cost functional method by extending it to the riccati equation setup our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive definite stochastic lq problems with such bridge some solvability relation between the indefinite and positive definite riccati equations is further characterized it is remarkable the solvability of the former is rather complicated than the latter hence our relation provides some alternative but useful viewpoint consequently the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the riccati equation in addition some example is studied using our theoretical results

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

stochastic maximum principle for delayed doubly stochastic control systems and their applications in this paper we investigate the optimal control problems for delayed doubly stochastic control systems we first discuss the existence and uniqueness of the delayed doubly stochastic differential

on stabilization by state feedback for neutral differential difference equations in this paper we consider a class of feedback operators for a linear differential difference equation of the neutral type we show that if a system has unstable neutral root chains any stabilizing feedback operator in this class must contain a derivative of the delayed state component under the assumption of exact controllability we construct a stabilizing feedback operator

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 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

stochastic neutral evolution equations on hilbert spaces with partially observed relaxed control and their necessary conditions of optimality abstract in this paper we develop the necessary conditions of optimality for partially observed control problems governed by stochastic neutral evolution equations on hilbert spaces driven by relaxed controls

risk sensitive zero sum differential games we consider two player risk sensitive zero sum differential games rszsdgs in our problem setup both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players and the objective functional is of the risk sensitive type first a stochastic maximum principle type necessary condition for an open loop saddle point of the rszsdg is established via nonlinear transformations of the adjoint processes of the equivalent risk neutral stochastic zero sum differential game in particular we obtain two variational inequalities namely the pair of saddle point inequalities of the rszsdg next we obtain the hamilton jacobi isaacs partial differential equation for the rszsdg which provides a sufficient condition for a feedback saddle point of the rszsdg using a logarithmic transformation of the associated value function finally we study the extended linear quadratic rszsdg lq rszsdg we show intractability of the extended lq rszsdg with the state and or controls of both players appearing in the diffusion term this unexpected intractability could lead to nonlinear open loop and feedback saddle points even if the problem itself is essentially lq and the isaacs condition holds

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

complex geodesics and iterates of holomorphic maps on convex domains in ⁿ we study complex geodesics f where is the unit disk in c and belongs to a class of bounded convex domains in c n with no boundary regularity assumption along with continuity up to the boundary existence of such complex geodesics with two prescribed values z w is established as a consequence we obtain some new results from iteration theory of holomorphic self maps of bounded convex domains in c n

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

large time behavior of a linear difference equation with rationally non related delays abstract we consider the linear difference equation x t j 1 n a j x t r j t 0 where a j 0 1 j n and the delays 0 r 1 r 2 r n are not rationally related in the sense that r j r k is irrational for some j and k it is shown that the large time behavior of the continuous solutions can be described in terms of the unique real root of the associated characteristic equation the proof is based on newman s tauberian theorem

on well posedness of singular neutral equations in the state space c abstract in this paper we study necessary und sufficient conditions for well posedness of scalar singular i e with a difference operator which is nonatomic at zero neutral functional differential equations snfdes in the state space c r 0 we also discuss an ill posed snfde in order to illustrate how the addition of a singularity away from zero in our case we have a perturbing atom at r can destroy well posedness

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

a kind of lq non zero sum differential game of backward stochastic differential equation with asymmetric information abstract this paper focuses on a kind of lq non zero sum differential game driven by backward stochastic differential equation with asymmetric information which is a natural continuation of wang and yu 2010 wang and yu 2012 different from wang and yu 2010 and wang and yu 2012 a realistic motivation for studying this kind of game is provided and some feedback nash equilibrium points are uniquely obtained by forward backward stochastic differential equations their filters and the corresponding riccati equations with markovian setting

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

a 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

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

stochastic maximum principle in the pontryagin s form for wide band noise driven systems the control problems for wide band noise driven systems have the following specific feature a wide band noise is given by its autocovariance function at the same time different wide band noises may have the same autocovariance function therefore it is important to obtain the results which are invariant under the change of wide band noises corresponding to the same autocovariance function in this paper one such result is obtained a stochastic maximum principle in the pontryagin s form is proved for a control system driven by a state dependent wide band noise the result is demonstrated on an example of controlled stochastic differential equation of risky asset price

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

on the function space controllability of linear neutral systems a criterion is derived for approximate controllability of linear autonomous neutral functional differential equations in the sobolev space w 2 1 h 0 mathbb e n controllability conditions are based on an abstract evolution equation representation of the system an abstract criterion applicable to a general class of neutral systems is obtained and for the special case of matrix systems of the form frac d dt left x t a 1 x t h right e 0 x t e 1 x t h bu t this abstract criterion leads to an algebraic criterion

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

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

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

stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and driven by relaxed controls in this paper we consider nonconvex control problems of stochastic differential equations driven by relaxed controls adapted in the weak star sense to a current of sigma algebras generated by observable processes we cover in a unified way both continuous diffusion and jump processes we present existence of optimal controls before we construct the necessary conditions of optimality unlike some papers in this area using only functional analysis we develop a stochastic hamiltonian system of equations on a rigorous basis using the semimartingale representation theory and the riesz representation theorem leading naturally to the existence of the adjoint process which satisfies a backward stochastic differential equation in other words our approach predicts the existence of the adjoint process as a natural consequence of riesz representation theory ensuring at the same time the weak star measurability this is unlike other papers where the adjoint process is introduced before its existence is prov

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

sensitivity analysis using ito malliavin calculus and martingales and application to stochastic optimal control we consider a multidimensional diffusion process x alpha t 0 leq t leq t whose dynamics depends on a parameter alpha our first purpose is to write as an expectation the sensitivity nabla alpha j alpha for the expected cost j alpha mathbb e f x alpha t in order to evaluate it using monte carlo simulations this issue arises for example from stochastic control problems where the controller is parameterized which reduces the control problem to a parametric optimization one or from model misspecifications in finance previous evaluations of nabla alpha j alpha using simulations were limited to smooth cost functions f or to diffusion coefficients not depending on alpha see yang and kushner siam j control optim 29 1991 pp 1216 1249 in this paper we cover the general case deriving three new approaches to evaluate nabla alpha j alpha which we call the malliavin calculus approach the adjoint approach and the martingale approach to accomplish this we leverage ito calculus malliavin calculus and martingale arguments in the second part of this work we provide discretization procedures to simulate the relevant random variables then we analyze their respective errors this analysis proves that the discretization error is essentially linear with respect to the time step this result which was already known in some specific situations appears to be true in this much wider context finally we provide numerical experiments in random mechanics and finance and compare the different methods in terms of variance complexity computational time and time discretization error

maximum principle for infinite horizon optimal control of mean field backward stochastic systems with delay and noisy memory in this paper we consider a problem of optimal control of an infinite horizon mean field backward stochastic differential equation with delay and noisy memory under partial information we derive

large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order the spectral theory for linear autonomous neutral functional differential equations fde yields explicit formulas for the large time behaviour of solutions our results are based on resolvent computations and dunford calculus applied to establish explicit formulas for the large time behaviour of solutions of fde we investigate in detail a class of two dimensional systems of fde

optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations abstract we study the boundary control problems for stochastic parabolic equations with neumann boundary conditions imposing super parabolic conditions we establish the existence and uniqueness of the solution of state and adjoint equations with non homogeneous boundary conditions by the galerkin approximations method we also find that in this case the adjoint equation bspde has two boundary conditions one is non homogeneous the other is homogeneous by these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method

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

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

linear multistep schemes for bsdes we study the convergence rate of a class of linear multistep methods for backward stochastic differential equations bsdes we show that under a sufficient condition on the coefficients the schemes enjoy a fundamental stability property coupling this result to an analysis of the truncation error allows us to design approximation with arbitrary order of convergence contrary to the analysis performed in w zhao g zhang and l ju siam j numer anal 48 2010 pp 1369 1394 we consider general diffusion models and bsdes with driver depending on z the class of methods we consider contains well known methods from the ode framework as nystrom milne or adams methods finally we provide a numerical illustration of the convergence of some methods

a generalized neyman pearson lemma for g probabilities this paper is concerned with hypothesis tests for g probabilities a class of nonlinear probability measures the problem is shown to be a special case of a general stochastic optimization problem where the objective is to choose the terminal state of certain backward stochastic differential equations so as to minimize a g expectation the latter is solved with a stochastic maximum principle approach neyman pearson type results are thereby derived for the original problem with both simple and randomized tests it turns out that the likelihood ratio in the optimal tests is nothing else than the ratio of the adjoint processes associated with the maximum principle concrete examples ranging from the classical simple tests financial market modelling with ambiguity to super and sub pricing of contingent claims and to risk measures are presented to illustrate the applications of the results obtained

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

characteristic matrices and spectral properties of evolutionary systems in this paper we introduce the notion of a characteristic matrix for a large class of unbounded operators and study the precise connection between characteristic matrices and spectral properties of evolutionary systems in particular we study so called multiplicity theorems several examples will illustrate our results

optimal bilinear control of stochastic nonlinear schrodinger equations mass sub critical case we study optimal control problems for stochastic nonlinear schrodinger equations in both the mass subcritical and critical case for general initial data of the minimal l 2 regularity we prove the existence and first order lagrange condition of an open loop control in particular these results apply to the stochastic nonlinear schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two respectively furthermore we obtain uniform estimates of backward stochastic solutions in new spaces of type u 2 and v 2 adapted to evolution operators related to linear schrodinger equations with lower order perturbations these estimates yield a new temporal regularity of backward stochastic solutions which is crucial for the tightness of approximating controls induced by ekeland s variational principle

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

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

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

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

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

the maximum principle for partially observed optimal control of stochastic differential equations this paper concerns partially observed optimal control of possibly degenerate stochastic differential equations with correlated noises between the system and the observation the control is allowed to enter into all the coefficients a general maximum principle is proved for the partially observed optimal control and the relations among the adjoint processes are established adjoint vector fields which are adapted to the past and present observations are introduced as the solutions to some backward stochastic partial differential equations bspdes and their relations are established under suitable conditions the adjoint processes are characterized in terms of the adjoint vector fields their differentials and hessians along the optimal state process some other formulations of the partially observed stochastic maximum principle are then derived

a new kind of accurate numerical method for backward stochastic differential equations in this paper we propose a new kind of numerical simulation method for backward stochastic differential equations bsdes we discretize the continuous bsdes on time space discrete grids use the monte carlo method to approximate mathematical expectations and use space interpolations to compute values at non grid points to demonstrate the accuracy and the effectiveness of our method several numerical examples are given

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

second order backward stochastic differential equations and fully nonlinear parabolic pdes for a d dimensional diffusion of the form dxt xt dt xt dwt and continuous functions f and g we study the existence and uniqueness of adapted processes y z and a solving the second order backward stochastic differential equation 2bsde dyt f t xt yt zt t dt z dxt t 0 t dzt at dt t dxt