In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwi...In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls U can be chosen to be non-convex but now the space of controls U can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.展开更多
The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems.The control domain is not necessarily convex and the cost functional can have a quadratic growth.In pa...The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems.The control domain is not necessarily convex and the cost functional can have a quadratic growth.In particular,they give a stochastic maximum principle for the linear quadratic optimal control problem.展开更多
The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The res...The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.展开更多
We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we...We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we obtain the stochastic maximum principle as the necessary condition for an optimal control,and we also prove its sufficiency under proper conditions.The stochastic linear quadratic problem in this setting is also discussed.展开更多
In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible ...In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland's variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.展开更多
This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s...This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.展开更多
An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to...An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.展开更多
This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state con...This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state constraint here is of non-functional type. The author puts forward two ways to understand the target set and the variation set. Then under two kinds of finite-codimensional conditions, the stochastic maximum principles are established, respectively. The main results are proved in two different ways. For the former, separating hyperplane method is used; for the latter, Ekeland＇s variational principle is applied. At last, the author takes the mean-variance portfolio selection with the box-constraint on strategies as an example to show the application in finance.展开更多
The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a ...The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a family of pairwise strongly orthonormal martingales associated with L6vy processes （see e.g., Nualart and Schoutens＇ paper in 2000）. We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria （or backward linear-quadratic problem, or BLQ problem for short） is discussed and characterized by a stochastic Hamilton system.展开更多
In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stoc...In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.展开更多
Different from the standard linear quadratic(LQ)problem for stochastic systems,the LQ problem considered in the paper has extra measurability restrictions.The problem also appears in the LQ control problem for stochas...Different from the standard linear quadratic(LQ)problem for stochastic systems,the LQ problem considered in the paper has extra measurability restrictions.The problem also appears in the LQ control problem for stochastic systems with delays,rational expectations problems,asymmetric information control,and so on.The essential difficulty lies in that one has to optimize the input and its conditional expectations simultaneously.The stochastic maximum principle(SMP)and orthogonal decomposition technique are the key tools.Firstly,the authors establish the SMP and convert the original problem into forward and backward stochastic difference equations(FBSDEs)with extra measurability restrictions.Secondly,the authors resolve the FBSDEs by using the orthogonal decomposition technique and obtain the analytical solution to the underlying problem.Thirdly,the authors explore the essential distinction between the problem and the standard stochastic LQ control problem.Finally,numerical examples are given to illustrate the obtained results.展开更多
This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales ass...This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.展开更多
This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type(MFSDEs in short),in which the coefficients depend on the state of...This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type(MFSDEs in short),in which the coefficients depend on the state of the solution process as well as of its expected value.Moreover,the cost functional is also of mean-field type.The control variable has two components,the first being absolutely continuous and the second singular.We establish necessary as well as sufficient conditions for optimal singular stochastic control where the system evolves according to MFSDEs.These conditions of optimality differs from the classical one in the sense that here the adjoint equation turns out to be a linear mean-field backward stochastic differential equation.The proof of our result is based on convex perturbation method of a given optimal control.The control domain is assumed to be convex.A linear quadratic stochastic optimal control problem of mean-field type is discussed as an illustrated example.展开更多
This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended t...This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.展开更多
The parameters of biological system may change under the influence of different states or state processes.The change of parameters can also affect the dynamic behaviors of epidemic disease.This paper presents a mean f...The parameters of biological system may change under the influence of different states or state processes.The change of parameters can also affect the dynamic behaviors of epidemic disease.This paper presents a mean field stochastic susceptible-infectedvaccinated(SIV)epidemic model which parameters depend on the state process.The sufficient and necessary conditions of optimal control of the novel epidemic model are obtained by maximum principle.Finally,we perform representative numerical simulations to illustrate the theoretical results and further discuss the feasibility based on the hand,foot and mouth disease(HFMD)data in China.展开更多
In this paper,we present a new precipitation model based on a multi-factor Ornstein-Uhlenbeck approach of pure-jump type.In this setup,we derive a representation for the related precipitation swap price process and in...In this paper,we present a new precipitation model based on a multi-factor Ornstein-Uhlenbeck approach of pure-jump type.In this setup,we derive a representation for the related precipitation swap price process and infer its risk-neutral time dynamics.We further deduce a pricing formula for European options written on the prccipitation swap and obtain the minimal variance hedging portfolio in the underlying weather market.In the second part of the paper,we provide a precipitation swap price representation under future information modeled by an initially enlarged filtration.We finally derive a formula for the associated information premium and investigate minimal variance hedging of prccipitation dcrivatives undcr futurc information.展开更多
In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence o...In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.展开更多
In this paper,the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way.They assume that the cost function satisfies a c...In this paper,the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way.They assume that the cost function satisfies a convexity and a weak monotonicity property.They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation(FBSDE for short).They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small,or when the convexity parameter of the running cost on the control is sufficiently large.Two different methods are developed.The first method is based on a continuation of the coefficients,which is developed for FBSDE by[Hu,Y.and Peng,S.,Solution of forward-backward stochastic differential equations,Probab.Theory Rel.,103(2),1995,273–283].They apply the method to conditional distribution dependent FBSDE.The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.展开更多
In this paper,we consider optimal control of stochastic differential equations subject to an expected path constraint.The stochastic maximum principle is given for a general optimal stochastic control in terms of cons...In this paper,we consider optimal control of stochastic differential equations subject to an expected path constraint.The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs.In particular,the compensated process in our adjoint equation is deterministic,which seems to be new in the literature.For the typical case of linear stochastic systems and quadratic cost functionals(i.e.,the so-called LQ optimal stochastic control),a verification theorem is established,and the existence and uniqueness of the constrained reflected FBSDEs are also given.展开更多
文摘In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls U can be chosen to be non-convex but now the space of controls U can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.
基金supported by the National Natural Science Foundation of China(No.12031009)。
文摘The authors give a stochastic maximum principle for square-integrable optimal control of linear stochastic systems.The control domain is not necessarily convex and the cost functional can have a quadratic growth.In particular,they give a stochastic maximum principle for the linear quadratic optimal control problem.
基金supported by the National Natural Science Foundation of China(No.61573217)the 111 Project(No.B12023)the National High-level Personnel of Special Support Program and the Chang Jiang Scholar Program of the Ministry of Education of China
文摘The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.
基金The authors are also grateful to the two anonymous referees for their valuable comments.J.Song is partially supported by Shandong University(Grant No.11140089963041)the National Natural Science Foundation of China(Grant No.12071256).
文摘We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter.Assuming the convexity of the control domain,we obtain the stochastic maximum principle as the necessary condition for an optimal control,and we also prove its sufficiency under proper conditions.The stochastic linear quadratic problem in this setting is also discussed.
基金supported by National Natural Science Foundation of China(Grant No.11401091)Postdoctoral Scientific Research Project of Jilin Province(Grant No.RB201357)+2 种基金the Fundamental Research Funds for the Central Universities(Grant No.14QNJJ002)China Postdoctoral Science Foundation(Grant No.2014M551152)the China Scholarship Council
文摘In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland's variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.
基金the financial support provided by the Swedish Research Council grant(2020-04697)the Norwegian Research Council grant(250768/F20),respectively。
文摘This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.
基金Supported by the National Natural Science Foundation of China(11101140,11301177)the China Postdoctoral Science Foundation(2011M500721,2012T50391)the Zhejiang Natural Science Foundation of China(Y6110775,Y6110789)
文摘An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.
基金supported by the National Natural Science Foundation of China under Grant No.11171076Science and Technology Commission,Shanghai Municipality under Grant No.14XD1400400
文摘This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state constraint here is of non-functional type. The author puts forward two ways to understand the target set and the variation set. Then under two kinds of finite-codimensional conditions, the stochastic maximum principles are established, respectively. The main results are proved in two different ways. For the former, separating hyperplane method is used; for the latter, Ekeland＇s variational principle is applied. At last, the author takes the mean-variance portfolio selection with the box-constraint on strategies as an example to show the application in finance.
基金supported by National Natural Science Foundation of China (Grant No. 11101090, 11101140, 10771122)Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090071120002)+2 种基金Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant No. T200924)Natural Science Foundation of Zhejiang Province (Grant No. Y6110775, Y6110789)Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry
文摘The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a family of pairwise strongly orthonormal martingales associated with L6vy processes （see e.g., Nualart and Schoutens＇ paper in 2000）. We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria （or backward linear-quadratic problem, or BLQ problem for short） is discussed and characterized by a stochastic Hamilton system.
文摘In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.
基金supported by the Original Exploratory Program Project of National Natural Science Foundation of China under Grant No.62250056the Joint Funds of the National Natural Science Foundation of China under Grant No.U23A20325+1 种基金the Major Basic Research of Natural Science Foundation of Shandong Province under Grant No.ZR2021ZD14the High-level Talent Team Project of Qingdao West Coast New Area under Grant No.RCTD-JC-2019-05。
文摘Different from the standard linear quadratic(LQ)problem for stochastic systems,the LQ problem considered in the paper has extra measurability restrictions.The problem also appears in the LQ control problem for stochastic systems with delays,rational expectations problems,asymmetric information control,and so on.The essential difficulty lies in that one has to optimize the input and its conditional expectations simultaneously.The stochastic maximum principle(SMP)and orthogonal decomposition technique are the key tools.Firstly,the authors establish the SMP and convert the original problem into forward and backward stochastic difference equations(FBSDEs)with extra measurability restrictions.Secondly,the authors resolve the FBSDEs by using the orthogonal decomposition technique and obtain the analytical solution to the underlying problem.Thirdly,the authors explore the essential distinction between the problem and the standard stochastic LQ control problem.Finally,numerical examples are given to illustrate the obtained results.
基金supported by the Major Basic Research Program of Natural Science Foundation of Shandong Province under Grant No.2019A01the Natural Science Foundation of Shandong Province of China under Grant No.ZR2020MF062。
文摘This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.
基金The authorwould like to thank the editor,the associate editor,and anonymous referees for their constructive corrections and valuable suggestions that improved the manuscript.The author was partially supported by Algerian PNR Project Grant 08/u07/857,ATRST-(ANDRU)2011-2013.
文摘This paper studies singular optimal control problems for systems described by nonlinear-controlled stochastic differential equations of mean-field type(MFSDEs in short),in which the coefficients depend on the state of the solution process as well as of its expected value.Moreover,the cost functional is also of mean-field type.The control variable has two components,the first being absolutely continuous and the second singular.We establish necessary as well as sufficient conditions for optimal singular stochastic control where the system evolves according to MFSDEs.These conditions of optimality differs from the classical one in the sense that here the adjoint equation turns out to be a linear mean-field backward stochastic differential equation.The proof of our result is based on convex perturbation method of a given optimal control.The control domain is assumed to be convex.A linear quadratic stochastic optimal control problem of mean-field type is discussed as an illustrated example.
基金supported by Science Engineering Research Board(SERB),DST,GovtYSS Project F.No:YSS/2014/000447 dated 20.11.2015UGC,New Delhi,for providing BSR fellowship for the year 2015.
文摘This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.
文摘The parameters of biological system may change under the influence of different states or state processes.The change of parameters can also affect the dynamic behaviors of epidemic disease.This paper presents a mean field stochastic susceptible-infectedvaccinated(SIV)epidemic model which parameters depend on the state process.The sufficient and necessary conditions of optimal control of the novel epidemic model are obtained by maximum principle.Finally,we perform representative numerical simulations to illustrate the theoretical results and further discuss the feasibility based on the hand,foot and mouth disease(HFMD)data in China.
文摘In this paper,we present a new precipitation model based on a multi-factor Ornstein-Uhlenbeck approach of pure-jump type.In this setup,we derive a representation for the related precipitation swap price process and infer its risk-neutral time dynamics.We further deduce a pricing formula for European options written on the prccipitation swap and obtain the minimal variance hedging portfolio in the underlying weather market.In the second part of the paper,we provide a precipitation swap price representation under future information modeled by an initially enlarged filtration.We finally derive a formula for the associated information premium and investigate minimal variance hedging of prccipitation dcrivatives undcr futurc information.
基金The research leading to these results received funding from the European Research Council under the European Community’s Seventh Framework Program(FP7/2007-2013)/ERC grant agreement 228087.
文摘In this paper,we study strongly robust optimal control problems under volatility uncertainty.In the G-framework,we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.
基金supported by the National Natural Science Foundation of China(No.12031009)。
文摘In this paper,the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way.They assume that the cost function satisfies a convexity and a weak monotonicity property.They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation(FBSDE for short).They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small,or when the convexity parameter of the running cost on the control is sufficiently large.Two different methods are developed.The first method is based on a continuation of the coefficients,which is developed for FBSDE by[Hu,Y.and Peng,S.,Solution of forward-backward stochastic differential equations,Probab.Theory Rel.,103(2),1995,273–283].They apply the method to conditional distribution dependent FBSDE.The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.
基金Ying Hu is partially supported by Lebesgue Center of Mathematics“Investissements d’avenir”Program(Grant No.ANR-11-LABX-0020-01)ANR CAESARS(Grant No.ANR-15-CE05-0024)+6 种基金ANR MFG(Grant No.ANR-16-CE40-0015-01)Shanjian Tang is partially supported by the National Science Foundation of China(Grant Nos.11631004 and 12031009)Zuo Quan Xu is partially supported by NSFC(Grant No.11971409)Research Grants Council of Hong Kong(GRF,Grant No.15202421)PolyU-SDU Joint Research Center on Financial MathematicsCAS AMSS-POLYU Joint Laboratory of Applied MathematicsHong Kong Polytechnic University.
文摘In this paper,we consider optimal control of stochastic differential equations subject to an expected path constraint.The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs.In particular,the compensated process in our adjoint equation is deterministic,which seems to be new in the literature.For the typical case of linear stochastic systems and quadratic cost functionals(i.e.,the so-called LQ optimal stochastic control),a verification theorem is established,and the existence and uniqueness of the constrained reflected FBSDEs are also given.
