regularity of the solution of darcy forchheimer s equation on considere l equation de darcy forchheimer v v v f div v 0 v 0 on demontre un theoreme d existence et d unicite et on etudie la regularite de la solution

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

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

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

the virtual element method for discrete fracture network simulations abstract in this work an optimization based approach presented in berrone et al 2013 2014 10 12 for discrete fracture network simulations is coupled with the virtual element method vem for the space discretization of the underlying darcy law the great flexibility of the vem in handling rather general polygonal elements allows in a natural way for an effective description of irregular solutions starting from an arbitrary triangulation which is built independently of the mesh on other fractures only partial conformity is in fact obtained with this approach numerical results performed on several dfn configurations confirm the viability and efficiency of the resulting method

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

modeling fractures and barriers as interfaces for flow in porous media we consider a fractured porous medium that is studied at a scale such that the fractures can be modeled individually models for flow in which the fractures are interfaces between subdomains are presented these models take into account interactions between the fractures and the surrounding porous medium existence and uniqueness of the solution to the model problem are proved error estimates show convergence as mathcal o max h d in the l2 norm where h is the mesh size and d the fracture width numerical experiments confirm the theoretical results

generalized reflected bsdes driven by a levy process and an obstacle problem for pdies with a nonlinear neumann boundary condition in this paper we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equations grbsdes in short driven by a levy process which involve the integral with respect to a continuous process by means of the snell envelope the penalization method and the fixed point theorem in addition we obtain the comparison theorem for the solutions of the grbsdes as an application we give a probabilistic formula for the viscosity solution of an obstacle problem for a class of partial differential integral equations pdies in short with a nonlinear neumann boundary condition

numerical discretization of a darcy forchheimer model we solve a steady darcy forchheimer flow in a bounded region by means of piecewise constant velocities and nonconforming piecewise mathbb p 1 pressures for the computation we solve the nonlinearity by an alternating directions algorithm and we decouple the computation of the velocity from that of the pressure by a gradient algorithm we prove a priori error estimates of the scheme and convergence of the alternating directions algorithm

discrete dynamical systems in l topological spaces in this paper we investigate iteration of zadeh image operators and their discrete dynamical systems in l topological spaces some basic concepts such as limit set invariant set periodic point nonwandering point and recurrent point are discussed in weakly induced topological spaces we also study topological mixing topological transitivity and minimality for such dynamical systems while comparing with those corresponding ones in crisp spaces

reflected diffusion processes with jumps a stochastic differential equation of wiener poisson type is considered in a d dimensional bounded region by using a penalization argument on the domain we are able to prove the existence and uniqueness of solutions in the strong sense the main assumptions are lipschitzian coefficients either convex or smooth domains and a regular outward reflecting direction as a direct consequence it is verified that the reflected diffusion process with jumps depends on the initial date in a lipschitz fashion

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

generalized bsdes and nonlinear neumann boundary value problems we study a new class of backward stochastic differential equations which involves the integral with respect to a continuous increasing process this allows us to give a probabilistic formula for solutions of semilinear partial differential equations with neumann boundary condition where the boundary condition itself is nonlinear we consider both parabolic and elliptic equations

backward doubly stochastic differential equations and systems of quasilinear spdes we introduce a new class of backward stochastic differential equations which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations thus extending the feynman kac formula for linear spde s

a generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks the simulation of flow in fractured media requires handling both a large number of fractures and a complex interconnecting network of these fractures networks considered in this paper are three dimensional domains made up of two dimensional fractures intersecting each other and randomly generated due to the stochastic generation of fractures intersections can be highly intricate the numerical method must generate a mesh and define a discrete problem for any discrete fracture network dfn a first approach erhel de dreuzy and poirriez siam j sci comput 31 2009 pp 2688 2705 is to generate a conforming mesh and to apply a mixed hybrid finite element method however the resulting linear system becomes very large when the network contains many fractures hence a second approach pichot erhel and de dreuzy appl anal 89 2010 pp 1629 1643 is to generate a nonconforming mesh using an independent mesh generation for each fracture then a mortar technique applied to the mixed hybrid finite element method deals with the nonmatching grids when intersections do not cross or overlap pairwise mortar relations for each intersection are efficient pichot erhel and de dreuzy 2010 but for most random networks discretized intersections involve more than two fractures in this paper we design a new method generalizing the previous one and that is applicable for stochastic networks the main idea is to combine pairwise mortar relations with additional relations for the overlapping part this method still ensures the continuity of fluxes and heads and still yields a symmetric positive definite linear system numerical experiments show the efficiency of the method applied to complex stochastic fracture networks we also study numerical convergence when reducing the mesh step this method makes it easy to perform mesh optimization and appears to be a very promising tool to simulate flow in multiscale fracture networks

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

a two grid block centered finite difference method for darcy forchheimer flow in porous media a two grid block centered finite difference method is proposed for solving the two dimensional darcy forchheimer model describing non darcy flow in porous media to construct the two grid method we modify the original nonlinear elliptic operator of darcy forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter varepsilon by using the two grid method solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid optimal order error estimates for pressure and velocity in discrete l 2 norms are obtained some numerical examples are given to show the accuracy and efficiency of the presented method

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

solution of forward backward stochastic differential equations in this paper we study the existence and uniqueness of the solution to forward backward stochastic differential equations without the nondegeneracy condition for the forward equation under a certain monotonicity condition we prove the existence and uniqueness of the solution to forward backward stochastic differential equations

backward doubly stochastic differential equations with weak assumptions on the coefficients in this paper we deal with one dimensional backward doubly stochastic differential equations bdsdes we obtain existence theorems and comparison theorems for solutions of bdsdes with weak assumptions on the coefficients

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

neoclassical econometries the kernel the paper considers the problem of determining the parameters in static equality constrained neoclassical models frequently encountered in economic analysis these include the utility maximizing household subject to income and time constraints and the profit maximizing firm subject to a given technology the approach applies equally to the traditional regression based stochastic econometric model with additive normally distributed errors or to a deterministic analog where the distance between the actual data and model solutions for the agents decisions is minimized conventional econometric practice is extended in that closed form analytic solutions for the decision functions are not required and essentially arbitrary configurations of available data are permitted the computer is employed to jointly 1 solve the agents optimization problems 2 impute missing data and 3 determine whether there is sufficient information in the available data set to uniquely calibrate or estimate the model s parameters

a block centered finite difference method for the darcy forchheimer model a block centered finite difference scheme is introduced to solve the nonlinear darcy forchheimer equation in which the velocity and pressure can be approximated simultaneously the second order error estimates for both pressure and velocity are established on a nonuniform rectangular grid numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis

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

stochastic pdies and backward doubly stochastic differential equations driven by levy processes in this paper a new class of backward doubly stochastic differential equations driven by teugels martingales associated with a levy process satisfying some moment condition and an independent brownian motion is investigated we obtain the existence and uniqueness of solutions to these equations a probabilistic interpretation for solutions to a class of stochastic partial differential integral equations is given

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

imposing curvature restrictions on flexible functional forms abstract a general computational method for estimating the parameters of a flexible functional form subject to convexity quasi convexity concavity or quasi concavity at a point at several points or over a region is set forth and illustrated with an example

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

flexible functional forms and global curvature conditions empirically estimated flexible functional forms frequently fail to satisfy the appropriate theoretical curvature conditions lau and gallant and golub have worked out methods for imposing the appropriate curvature conditions locally but those local techniques frequently fail to yield satisfactory results we develop two methods for imposing curvature conditions globally in the context of cost function estimation the first method adopts lau s technique to a generalization of a functional form first proposed by mcfadden using this generalized mcfadden functional form it turns out that imposing the appropriate curvature conditions at one data point imposes the conditions globally the second method adopts a technique used by mcfadden and barnett which is based on the fact that a non negative sum of concave functions will be concave our various suggested techniques are illustrated using the u s manufacturing data utilized by berndt and khaled

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

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

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 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

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

geometric multigrid methods for darcy forchheimer flow in fractured porous media in this paper we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media specifically we consider a mixed dimensional model which couples darcy flow in the porous matrix with forchheimer flow within the fractures a suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure in order to solve this system we propose a full approximation scheme fas multigrid solver that appropriately deals with the mixed dimensional nature of the problem by using mixed dimensional smoothing and inter grid transfer operators numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability the forchheimer coefficient and the mesh size the case of several possibly intersecting fractures in a heterogeneous porous medium is also discussed

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

monolithic multigrid method for the coupled stokes flow and deformable porous medium system abstract the interaction between fluid flow and a deformable porous medium is a complicated multi physics problem which can be described by a coupled model based on the stokes and poroelastic equations a monolithic multigrid method together with either a coupled vanka smoother or a decoupled uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids a specialty in our modeling approach is that at the interface of the fluid and poroelastic medium two unknowns from the different subsystems are defined at the same grid point we propose a special discretization at and near the points on the interface which combines the approximation of the governing equations and the considered interface conditions in the decoupled uzawa smoother local fourier analysis lfa helps us to select optimal values of the relaxation parameter appearing to implement the monolithic multigrid method grid partitioning is used to deal with the interface updates when communication is required between two subdomains numerical experiments show that the proposed numerical method has an excellent convergence rate the efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients

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

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

backwards sde with random terminal time and applications to semilinear elliptic pde suppose im t is the filtration induced by a wiener process w in r d tau is a finite im t stopping time terminal time xi is an im tau measurable random variable in r k terminal value and f cdot y z is a coefficient process depending on y in r k and z in l r d r k satisfying y tilde y f s y z f s tilde y z leq a y tilde y 2 f need not be lipschitz in y and f s y z f s y tilde z leq b z tilde z for some real a and b plus other mild conditions we identify a hilbert space depending on tau and on the number gamma equiv b 2 2a in which there exists a unique pair of adapted processes y z satisfying the stochastic differential equation dy s 1 s leq tau z s dw s f s y s z s ds with the given terminal condition y tau xi provided a certain integrability condition holds this result is applied to construct a continuous viscosity solution to the dirichlet problem for a class of semilinear elliptic pde s

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

econometric methods for agricultural supply under uncertainty fertilizer use and crop response the paper develops general econometric methods for the treatment of modelling decision making under conditions of uncertainty and risk the problem of modelling fertilizer use and crop response in the context of irrigated and or rainfed water supply when there is uncertainty and risk with respect to weather conditions over the crop cycle and the subsequent price at marketing suggested the present approach as a by product the stochastic duality theory associated with the present primal problem is also derived

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

well posedness of fully coupled fracture bulk darcy flow with xfem in this work we consider the coupled problem of darcy flow in a fracture and the surrounding porous medium the fracture is represented as a d 1 dimensional interface and it is nonmatching with the computational grid thanks to a suitable extended finite element method xfem enrichment of the mixed finite element spaces in the existing literature well posedness has been proven for the discrete problem in the hypothesis of a given solution in the fracture this work provides theoretical results on the stability and convergence of the discrete fully coupled problem yielding sharp conditions on the fracture geometry and on the computational grid to ensure that the inf sup condition is satisfied by the enriched spaces as confirmed by numerical experiments

a multipoint flux mixed finite element method for the compressible darcy forchheimer models abstract we develop a multipoint flux mixed finite element method for the compressible darcy forchheimer models this method is motivated by the multipoint flux approximation method based on the lowest order brezzi douglas marini mixed finite element method and combined with a special quadrature rule that allows for local velocity elimination and leads to a cell centered system for the pressures we develop two approximation methods for the problems the semi discrete method and fully discrete method where the fully discrete method includes the backward euler scheme and the crank nicolson scheme theoretical results show the convergences for both pressure and velocity the numerical experiments show that the convergence rates of the method are in agreement with the theoretical analysis

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 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

uzawa smoother in multigrid for the coupled porous medium and stokes flow system the multigrid solution of coupled porous media and stokes flow problems is considered the darcy equation as the saturated porous medium model is coupled to the stokes equations by means of appropriate interface conditions we focus on an efficient multigrid solution technique for the coupled problem which is discretized by finite volumes on staggered grids giving rise to a saddle point linear system special treatment is required regarding the discretization at the interface an uzawa smoother is employed in multigrid which is a decoupled procedure based on symmetric gauss seidel smoothing for velocity components and a simple richardson iteration for the pressure field since a relaxation parameter is part of a richardson iteration local fourier analysis is applied to determine the optimal parameters highly satisfactory multigrid convergence is reported and moreover the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity which are relevant for applications

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

stationary solutions of spdes and infinite horizon bdsdes in this paper we study the existence of stationary solutions for stochastic partial differential equations we establish a new connection between l 2 rd r1 l 2 rd rd valued solutions of backward doubly stochastic differential equations bdsdes on infinite horizon and the stationary solutions of the spdes moreover we prove the existence and uniqueness of the solutions of bdsdes on both finite and infinite horizons so obtain the solutions of initial value problems and the stationary solutions independent of any initial value of spdes the connection of the weak solutions of spdes and bdsdes has independent interests in the areas of both spdes and bsdes

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

lp solutions of infinite time interval backward doubly stochastic differential equations under monotonicity and general increasing conditions abstract in this paper we study the existence and uniqueness theorem for l p 1 p 2 solutions to a class of infinite time interval backward doubly stochastic differential equations bdsdes for short under monotonicity and general increasing conditions furthermore we obtain the comparison theorem for 1 dimensional infinite time interval bdsdes in l p 1 p 2

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

neoclassical econometrics the agenda abstract a new approach to econometric modeling and parameter estimation which permits the construction of macro econometric models from general microeconomic foundations is proposed it provides inter alia full information calibration estimation methods which 1 a permits a statistical solution to the exact or consistent aggregation problem 2 b permits a new approach to the problem of parameter identification 3 c permits a systematic approach to the problem of model selection 4 d obviates the need for use of the duality theory of cost and profit functions by working solely on primal problems 5 e treats in principle the spectrum of perfectly competitive to monopolistic monopsonistic market structures 6 f applies in principle to both partial and general equilibrium models and 7 g obtains the structural form of limited dependent variable econometric estimation methods directly from neoclassical constrained optimization behavioral models all in the context of well behaved but essentially unrestricted functional forms subject to a general set of rationality postulates regarding the behavior of individual economic units

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

on a class of backward doubly stochastic differential equations in this paper a new class of backward doubly stochastic differential equations is studied this type of equations has a more general form of the forward ito integrals compared to the ones which have been studied until now we conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations considered earlier in paper 5 by pardoux and peng some comparison theorems are also given as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations

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

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation