reflected mean field backward stochastic differential equations approximation and associated nonlinear pdes abstract mathematical mean field approaches have been used in many fields not only in physics and chemistry but also recently in finance economics and game theory in this paper we will study a new special mean field problem in a purely probabilistic method to characterize its limit which is the solution of mean field backward stochastic differential equations bsdes with reflections on the other hand we will prove that this type of reflected mean field bsdes can also be obtained as the limit equation of the mean field bsdes by penalization method finally we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean field bsdes

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

optimal control of mean field backward doubly stochastic systems driven by ito levy processes abstractin this paper we introduce a new class of backward doubly stochastic differential equations in short bdsde called mean field backward doubly stochastic differential equations in short m

nontrivial solution for a semilinear elliptic equation in unbounded domain with critical sobolev exponent where is an unbounded domains with smooth boundary in r 2 2n n 2 a x c1 satisfies the following conditions a1 a ln 2 a2 x a x 0 such that b 4 when is a bounded domain with smooth boundary in r n 5 similar problem has been studied by many mathematicians for example in 3 brezis and nirenberg studied problem p in 4 capozzi et al prove that when

stochastic maximum principle for optimal control problems of forward backward systems involving impulse controls we consider a stochastic optimal control problem of a forward backward system in which the control variable consists of two components the continuous control and the impulse control the domain of the control is assumed to be convex necessary optimality conditions of the pontryagin maximum principle type are obtained for this stochastic optimal control problem we also give additional conditions under which the necessary optimality conditions turn out to be sufficient

indefinite mean field stochastic linear quadratic optimal control this paper is concerned with the discrete time indefinite mean field linear quadratic optimal control problem the so called mean field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals such as the mean variance portfolio selection problems a dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage wise backward recursion the classical dynamic programming based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies in this paper we show that both the well posedness and the solvability of the indefinite mean field linear quadratic problem are equivalent to the solvability of two coupled constrained generalized difference riccati equations and a constrained linear recursive equation we characterize the optimal control set completely and obtain a set of necessary and sufficient conditions on the mean variance portfolio selection problem the results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean variance type portfolio selection problems

nonnegative primitive matrices with exponent 2 abstract a nonnegative matrix m with zero trace is primitive if for some positive integer k m k is positive the exponent exp m of the primitive matrix is the smallest such k by treating the digraph g whose adjacency matrix is the primitive matrix m we will show that the minimum number of positive entries of m is 3 n 3 when exp m 2 we will also show that for a symmetric n n matrix m if exp m 2 the minimum number of positive entries of m is 3 n 2 or 3 n 3 according to n

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

stabilization control for linear continuous time mean field systems this paper investigates the stabilization and control problems for linear continuous time mean field systems under standard assumptions the necessary and sufficient conditions to stabilize the mean field systems in the mean square sense are explored for the first time it is shown that under the assumption of exact detectability exact observability the mean field system is stabilizable if and only if a coupled algebraic riccati equation admits a unique positive semidefinite solution positive definite solution which coincides with the classical stabilization results for standard deterministic systems and stochastic systems

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

primitive graphs with given exponents and minimum number of edges a graph g v e on n vertices is primitive if there is a positive integer k such that for each pair of vertices u v of g there is a walk of length k from u to v the minimum value of such an integer k is the exponent exp g of g in this paper we find the minimum number h n k of edges of a simple graph g on n vertices with exponent k and we characterize all graphs which have h n k edges when k is 3 or even

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

a stochastic maximum principle for systems with jumps with applications to finance abstract we consider a stochastic control problem with linear dynamics with jumps convex cost criterion and convex state constraint in which the control enters the drift the diffusion and the jump coefficients we allow these coefficients to be random and do not impose any l p bounds on the control we obtain a stochastic maximum principle for this model that provides both necessary and sufficient conditions of optimality this is the first version of the stochastic maximum principle that covers the consumption investment problem in which there are jumps in the price system

mean field stochastic h 2 h control with delay the finite horizon h 2 h control problem of mean field type for delayed stochastic systems is considered in this paper firstly we derive a sufficient mean field generalised stochastic bounded

risk sensitive mean field type games with l p norm drifts we study how risk sensitive players act in situations where the outcome is influenced not only by the state action profile but also by the distribution of it in such interactive decision making problems the classical mean field game framework does not apply we depart from most of the mean field games literature by presuming that a decision maker may include its own state distribution in its decision this leads to the class of mean field type games in mean field type situations a single decision maker may have a big impact on the mean field terms for which new type of optimality equations are derived we establish a finite dimensional stochastic maximum principle for mean field type games where the drift functions have a p norm structure which weaken the classical lipschitz and differentiability assumptions sufficient optimality equations are established via dynamic programming principle but in infinite dimension using de finetti hewitt savage theorem we show that a propagation of chaos property with virtual particles holds for the non linear mckean vlasov dynamics

the characterization of symmetric primitive matrices with exponent n 1 abstract the symmetric primitive matrices with exponent n has been described completely in linear multilinear algebra 39 1995 391 in this paper the complete characterization of symmetric primitive matrices with exponent n 1 is obtained

discrete time mean field stochastic linear quadratic optimal control problems this paper firstly presents necessary and sufficient conditions for the solvability of discrete time mean field stochastic linear quadratic optimal control problems secondly the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method thirdly by introducing several sequences of bounded linear operators the problem is formulated as an operator stochastic linear quadratic optimal control problem by the kernel range decomposition representation of the expectation operator and its pseudo inverse the optimal control is derived using solutions to two algebraic riccati difference equations finally by completing the square the two riccati equations and the optimal control are also obtained

mean field stochastic linear quadratic optimal control with markov jump parameters abstract this paper considers a class of mean field stochastic linear quadratic optimal control problems with markov jump parameters the new feature of these problems is that means of state and control are incorporated into the systems and the cost functional based on the modes of markov chain the corresponding decomposition technique of augmented state and control is introduced it is shown that under some appropriate conditions there exists a unique optimal control which can be explicitly given via solutions of two generalized difference riccati equations a numerical example sheds light on the theoretical results established

necessary and sufficient optimality conditions for relaxed and strict control problems we consider a stochastic control problem where the set of strict classical controls is not necessarily convex and the system is governed by a nonlinear stochastic differential equation in which the control enters both the drift and the diffusion coefficients by introducing a new approach we establish necessary as well as sufficient conditions of optimality for two models the first concerns the relaxed controls which are measure valued processes in which an optimal solution exists the second is a particular case of the first and relates to strict control problems these results are given in the form of global stochastic maximum principle by using only the first order expansion and the associated adjoint equation this improves all of the previous works on the subject

wielandt type theorem for cartesian product of digraphs abstract we show that mn 1 is an upper bound of the exponent of the cartesian product d e of two digraphs d and e on m n vertices respectively and we prove our upper bound is extremal when m n 1 we also find all d and e when the exponent of d e is mn 1 in addition when m n we prove that the extremal upper bound of exp d e is n 2 n 1 and only the cartesian product z n w n of the directed cycle and wielandt digraph has exponent equals to this bound

optimality necessary conditions in singular stochastic control problems with nonsmooth data the present paper studies the stochastic maximum principle in singular optimal control where the state is governed by a stochastic differential equation with nonsmooth coefficients allowing both

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

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

the exponent set of symmetric primitive 0 1 matrices with zero trace abstract we prove that the exponent set of symmetric primitive 0 1 matrices with zero trace the adjacency matrices of the simple graphs is 2 3 2 n 4 s where s is the set of all odd numbers in n 2 n 1 2 n 5 we also obtain a characterization of the symmetric primitive matrices with zero trace whose exponents attain the upper bound 2 n 4

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

linear quadratic mean field stochastic zero sum differential games abstract in this paper we consider the two player linear quadratic mean field stochastic differential game lq mf szsdg where the expected values of state and players control variables are included in the corresponding mean field stochastic system and the objective functional we characterize the explicit condition under which the feedback saddle point equilibrium for the lq mf szsdg exists the results are obtained by using the completion of squares method and the ordered interchangeability property of multiple saddle points we show that the corresponding saddle point equilibrium is linear in state which is characterized by the two coupled riccati differential equations crdes we also discuss the solvability conditions of the crdes numerical results are presented to verify the theoretical results of the paper

existence of solutions for an elliptic equation with indefinite weight abstract we consider the elliptic equation with indefinite weight u v x u f x u u h 1 r n where v x is a function possibly changing sign in r n under certain assumptions on f x u we obtain the existence of m n pairs of nontrivial solutions where m n and the m and n are all integers

on lewin and vitek s conjecture about the exponent set of primitive matrices m lewin and y vitek conjecture that every integer w 1 i n2 2n 2 1 is the exponent of some n x n primitive matrix in this paper we prove that this conjecture is true except for n 11 the problem of determining the exponent set e is completely solved

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

mean field backward stochastic differential equations a limit approach mathematical mean field approaches play an important role in different fields of physics and chemistry but have found in recent works also their application in economics finance and game theory the objective of our paper is to investigate a special mean field problem in a purely stochastic approach for the solution y z of a mean field backward stochastic differential equation driven by a forward stochastic differential of mckean vlasov type with solution x we study a special approximation by the solution x n y n z n of some decoupled forward backward equation which coefficients are governed by n independent copies of x n y n z n we show that the convergence speed of this approximation is of order 1 root n moreover our special choice of the approximation allows to characterize the limit behavior of root n x n x y n y z n z we prove that this triplet converges in law to the solution of some forward backward stochastic differential equation of mean field type which is not only governed by a brownian motion but also by an independent gaussian field

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

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

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

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

the stochastic maximum principle for linear convex optimal control with random coefficients this paper considers a stochastic control problem with linear dynamics convex cost criterion and convex state constraint in which the control enters both the drift and diffusion coefficients these coefficients are allowed to be random and no l p bounds are imposed on the control an explicit solution for the adjoint equation and a global stochastic maximum principle are obtained for this model this is evidently the first version of the stochastic maximum principle that covers the consumption investment problem the mathematical tools are those of stochastic calculus and convex analysis when it is assumed as in other versions of the stochastic maximum principle that the admissible controls are square integrable not only a necessary but also a sufficient condition for optimality is obtained it is then shown that this particular case of the general model may be applied to solve a variety of problems in stochastic control including the linear regulator predicted miss and benes problems

on an electromagnetic schrodinger equation with critical growth abstract in this paper we consider the electromagnetic schrodinger problem i a 2 u v x u k x u 2 2 u in h a v 1 r n see the definition in section 1 according as whether the indefinite linear is resonant or nonresonant we structure different direct sums of h a v 1 r n then by applying the linking theorem twice to the corresponding functional we establish the existence results

the stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls this paper is concerned with a pontryagin s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls this kind of control problems is motivated by some interesting phenomena arising from economics and finance we establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis to explain the theoretical results we apply them to a production and consumption choice problem

an open loop stackelberg strategy for the linear quadratic mean field stochastic differential game this paper is concerned with the open loop linear quadratic lq stackelberg game of the mean field stochastic systems in finite horizon by means of two generalized differential riccati equations the follower first solves a mean field stochastic lq optimal control problem then the leader turns to solve an optimization problem for a linear mean field forward backward stochastic differential equation by introducing new state and costate variables we present a sufficient condition for the existence and uniqueness of the stackelberg strategy in terms of the solvability of some riccati equations and a convexity condition furthermore it is shown that the open loop stackelberg equilibrium admits a feedback representation involving the new state and its mean finally two examples are given to show the effectiveness of the proposed results

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

a stochastic maximum principle for delayed mean field stochastic differential equations and its applications in this technical note we discuss the stochastic optimal control problems of mean field stochastic differential delayed equations mfsddes which arise naturally from various backgrounds including economics finance engineering and physics etc to this end some new estimates are used to handle the complex structure of our controlled system due to the presence of both delay and mean field characters as the main result a stochastic maximum principle for the mean field stochastic optimal control with delay mfsocd is derived in terms of necessary and sufficient conditions in particular applying the convex variation and duality relation we obtain the necessary condition for optimality see theorem 1 in addition the sufficient condition of the optimality is also obtained under some convex condition see theorem 2 based on our maximum principle the related mean field linear quadratic delayed mflqd optimal control problems are also investigated the optimal control is derived and its existence is also verified refer theorem 3 as illustration an example is also proposed and its explicit optimal control is derived

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

optimality conditions for partial information stochastic control problems driven by levy processes abstract in this paper we consider a partial information stochastic control problem where the system is governed by a nonlinear stochastic differential equation driven by teugels martingales associated with some levy process and an independent brownian motion we prove optimality necessary conditions in the form of a maximum principle these conditions turn out to be sufficient under some convexity assumptions to illustrate the general results an example is solved

the maximum principle for a jump diffusion mean field model and its application to the mean variance problem abstract this paper establishes a necessary and sufficient stochastic maximum principle for a mean field model with randomness described by brownian motions and poisson jumps we also prove the existence and uniqueness of the solution to a jump diffusion mean field backward stochastic differential equation a new version of the sufficient stochastic maximum principle which only requires the terminal cost is convex in an expected sense is applied to solve a bicriteria mean variance portfolio selection problem

optimal control of mean field jump diffusion systems with noisy memory abstractthis paper is concerned with an optimal control problem under mean field jump diffusion systems with delay and noisy memory first we derive necessary and sufficient maximum principles using malliavin calculus technique meanwhile we introduce a new mean field backward stochastic differential equation as the adjoint equation which involves not only partial derivatives of the hamiltonian function but also their malliavin derivatives then applying a reduction of the noisy memory dynamics to a two dimensional discrete delay optimal control problem we establish the second set of necessary and sufficient maximum principles under partial information moreover a natural link between the above two approaches is established via the adjoint equations finally we apply our theoretical results to study a mean field linear quadratic optimal control problem

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

a stochastic maximum principle for a markov regime switching jump diffusion model and its application to finance this paper develops a sufficient stochastic maximum principle for a stochastic optimal control problem where the state process is governed by a continuous time markov regime switching jump diffusion model we also establish the relationship between the stochastic maximum principle and the dynamic programming principle in a markovian case applications of the stochastic maximum principle to the mean variance portfolio selection problem are discussed