A stochastic maximum principle for the risk-sensitive optimal control prob- lem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where ...A stochastic maximum principle for the risk-sensitive optimal control prob- lem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.展开更多
This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables...This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.展开更多
In this paper, we study the stochastic maximum principle for optimal control prob- lem of anticipated forward-backward system with delay and Lovy processes as the random dis- turbance. This control system can be descr...In this paper, we study the stochastic maximum principle for optimal control prob- lem of anticipated forward-backward system with delay and Lovy processes as the random dis- turbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and L^vy processes （AFBSDEDLs）, we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs＇ preliminary result with certain classical convex variational techniques, the corresponding maxi- mum principle is proved.展开更多
Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly p-local Dirichlet fo...Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly p-local Dirichlet form on the abstract metric measure space. As an application we obtain lower estimates for heat kernels on some Riemannian manifolds.展开更多
This paper considered the optimal control problem for distributed parameter systems with mixed phase-control constraints and end-point constraints. Pontryagin's maximum principle for optimal control are derived vi...This paper considered the optimal control problem for distributed parameter systems with mixed phase-control constraints and end-point constraints. Pontryagin's maximum principle for optimal control are derived via Duboviskij-Milujin theorem.展开更多
Optimal glucose feed strategy for glycerol fed-batch fermentation was investigated by Pontryagin’s maximum principle to maximize the final glycerol yield. The problem was solved by a nonsingular control approach by s...Optimal glucose feed strategy for glycerol fed-batch fermentation was investigated by Pontryagin’s maximum principle to maximize the final glycerol yield. The problem was solved by a nonsingular control approach by selecting the culture volume as the control variable, then the general optimal feed profile was numerically determined.展开更多
A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary an...A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2/H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.展开更多
In this note we announce the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi’s iteration.In particular,the existence of weak solutions for possibly degenerate s...In this note we announce the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi’s iteration.In particular,the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion coefficients is obtained.展开更多
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
In this present work,we consider an eigenvalue-type problem involving the p-Laplacian operator under nonlinear boundary conditions,in case g≡0,and we prove the existence of an isolated eigenvalue plus a continuous fa...In this present work,we consider an eigenvalue-type problem involving the p-Laplacian operator under nonlinear boundary conditions,in case g≡0,and we prove the existence of an isolated eigenvalue plus a continuous family of eigenvalues.We also study the maximum principle-type for(P_(λ,g)) with g≥0,g■0.展开更多
This paper is concerned with the ergodic stochastic optimal control problem with Markov Regime-Switching in a dissipative system.The proposed approach primarily relies on duality techniques.The control system is descr...This paper is concerned with the ergodic stochastic optimal control problem with Markov Regime-Switching in a dissipative system.The proposed approach primarily relies on duality techniques.The control system is described by controlled dissipative stochastic diferential equations and modulated by a continuous-time,fnite-state Markov chain.The cost functional is ergodic,which is the expected long-run mean average type.The control domain is assumed to be convex,and the convex variation technique is used.Both necessary condition version and sufcient condition version of the stochastic maximum principle are established for optimal control.An example is discussed to illustrate the signifcance of our results.展开更多
The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations(BSDEs).The system coefficients depend on parameter θ,whi...The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations(BSDEs).The system coefficients depend on parameter θ,while the generator of BSDEs exhibits quadratic growth with respect to z.To address the uncertainty present in the model,the variational inequality is derived using weak convergence techniques.Additionally,due to the generator being quadratic with respect to z,the forward adjoint equations are stochastic differential equations with unbounded coefficients,involving mean oscillation martingales.By using the reverse Holder inequality and John-Nirenberg inequality,we demonstrate that the solutions are continuous with respect to parameter θ.Moreover,the necessary and sufficient conditions for robust optimal control are established using the linearization method.展开更多
In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for th...In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.展开更多
Both necessary and sufficient maximum principles for optimal control of stochastic systemwith random jumps consisting of forward and backward state variables are proved.The control variableis allowed to enter both dif...Both necessary and sufficient maximum principles for optimal control of stochastic systemwith random jumps consisting of forward and backward state variables are proved.The control variableis allowed to enter both diffusion and jump coefficients.The result is applied to a mean-varianceportfolio selection mixed with a recursive utility functional optimization problem.Explicit expressionof the optimal portfolio selection strategy is obtained in the state feedback form.展开更多
It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which ...It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.展开更多
The paper is concerned with a stochastic optimal control problem in which the controlled system is described by a fully coupled nonlinear forward-backward stochastic differential equation driven by a Brownian motion.I...The paper is concerned with a stochastic optimal control problem in which the controlled system is described by a fully coupled nonlinear forward-backward stochastic differential equation driven by a Brownian motion.It is required that all admissible control processes are adapted to a given subfiltration of the filtration generated by the underlying Brownian motion.For this type of partial information control,one sufficient(a verification theorem) and one necessary conditions of optimality are proved.The control domain need to be convex and the forward diffusion coefficient of the system can contain the control variable.展开更多
This paper is concerned with the optimal control problems of forward-backward delay systems involving impulse controls. The authors establish a stochastic maximum principle for this kind of systems. The most distingui...This paper is concerned with the optimal control problems of forward-backward delay systems involving impulse controls. The authors establish a stochastic maximum principle for this kind of systems. The most distinguishing features of the proposed problem are that the control variables consist of regular and impulsive controls, both with time delay, and that the domain of regular control is not necessarily convex. The authors obtain the necessary and sufficient conditions for optimal controls,which have potential applications in mathematical finance.展开更多
This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backw...This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backward stochastic differential equation theory with certain classical convex variational techniques, the necessary maximum principle is proved for the partially observed optimal control, where the control domain is a nonempty convex set. Under certain convexity assumptions, the author also gives the sufficient conditions of an optimal control for the aforementioned optimal optimal problem. To illustrate the theoretical result, the author also works out an example of partial information linear-quadratic optimal control, and finds an explicit expression of the corresponding optimal control by applying the necessary and sufficient maximum principle.展开更多
In this paper,we study the recursive stochastic optimal control problems.The control domain does not need to be convex,and the generator of the backward stochastic differential equation can contain z.We obtain the var...In this paper,we study the recursive stochastic optimal control problems.The control domain does not need to be convex,and the generator of the backward stochastic differential equation can contain z.We obtain the variational equations for backward stochastic differential equations,and then obtain the maximum principle which solves completely Peng’s open problem.展开更多
基金supported by the National Basic Research Program of China (973 Program, 2007CB814904)the National Natural Science Foundations of China (10921101)+2 种基金Shandong Province (2008BS01024, ZR2010AQ004)the Science Funds for Distinguished Young Scholars of Shandong Province (JQ200801)Shandong University (2009JQ004),the Independent Innovation Foundations of Shandong University (IIFSDU,2009TS036, 2010TS060)
文摘A stochastic maximum principle for the risk-sensitive optimal control prob- lem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.
基金supported by the National Natural Science Foundation of China(11701214)Shandong Provincial Natural Science Foundation,China(ZR2019MA045).
文摘This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.
基金Supported by the National Natural Science Foundation(11221061 and 61174092)111 project(B12023),the National Science Fund for Distinguished Young Scholars of China(11125102)Youth Foundation of QiLu Normal Institute(2012L1010)
文摘In this paper, we study the stochastic maximum principle for optimal control prob- lem of anticipated forward-backward system with delay and Lovy processes as the random dis- turbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and L^vy processes （AFBSDEDLs）, we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs＇ preliminary result with certain classical convex variational techniques, the corresponding maxi- mum principle is proved.
文摘Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly p-local Dirichlet form on the abstract metric measure space. As an application we obtain lower estimates for heat kernels on some Riemannian manifolds.
文摘This paper considered the optimal control problem for distributed parameter systems with mixed phase-control constraints and end-point constraints. Pontryagin's maximum principle for optimal control are derived via Duboviskij-Milujin theorem.
基金From National Ninth Five Years Project (NO. 96-03-03-03A).
文摘Optimal glucose feed strategy for glycerol fed-batch fermentation was investigated by Pontryagin’s maximum principle to maximize the final glycerol yield. The problem was solved by a nonsingular control approach by selecting the culture volume as the control variable, then the general optimal feed profile was numerically determined.
基金supported by the Doctoral foundation of University of Jinan(XBS1213)the National Natural Science Foundation of China(11101242)
文摘A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2/H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.
基金The China Scholarship Council,the National Basic Research Program(2009CB219301) of China(973) in partthe National Public Benefit Scientific Research Foundation(201011078) of China+2 种基金the National Innovation Research Project for Exploration and Development of Oil Shale(OSP-02 and OSR-02)the NSF(41304087,11071026,61133011,61170092,60973088,61202308,11001100,11171131 and 11026043) of Chinathe Basic Research Foundation of Jilin University in 2012
文摘In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.
基金National Natural Science Foundation of China(11731009).
文摘In this note we announce the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi’s iteration.In particular,the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion coefficients is obtained.
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.
文摘In this present work,we consider an eigenvalue-type problem involving the p-Laplacian operator under nonlinear boundary conditions,in case g≡0,and we prove the existence of an isolated eigenvalue plus a continuous family of eigenvalues.We also study the maximum principle-type for(P_(λ,g)) with g≥0,g■0.
基金supported by the National Key Research and Development Program of China(No.2023YFA1009200)the National Natural Science Foundation of China(Nos.11831010,61961160732)+1 种基金the Shandong Provincial Natural Science Foundation(No.ZR2019ZD42)the Taishan Scholars Climbing Program of Shandong Grant(No.TSPD20210302)。
文摘This paper is concerned with the ergodic stochastic optimal control problem with Markov Regime-Switching in a dissipative system.The proposed approach primarily relies on duality techniques.The control system is described by controlled dissipative stochastic diferential equations and modulated by a continuous-time,fnite-state Markov chain.The cost functional is ergodic,which is the expected long-run mean average type.The control domain is assumed to be convex,and the convex variation technique is used.Both necessary condition version and sufcient condition version of the stochastic maximum principle are established for optimal control.An example is discussed to illustrate the signifcance of our results.
基金supported by the National Key R&D Program of China(Grant Nos.2022YFA1006101 and 2022YFA1006102)National Natural Science Foundation of China(Grant Nos.72171133,12101291,and 12371445)+3 种基金Natural Science Foundation of Shandong Province(Grant Nos.ZR2024MA039 and ZR2022MA029)Guangdong Basic and Applied Basic Research Foundation(Grant No.2025B1515020091)Shenzhen Fundamental Research General Program(Grant No.JCYJ20230807093309021)Science and Technology Commission of Shanghai Municipality(Grant No.22ZR1407600).
文摘The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations(BSDEs).The system coefficients depend on parameter θ,while the generator of BSDEs exhibits quadratic growth with respect to z.To address the uncertainty present in the model,the variational inequality is derived using weak convergence techniques.Additionally,due to the generator being quadratic with respect to z,the forward adjoint equations are stochastic differential equations with unbounded coefficients,involving mean oscillation martingales.By using the reverse Holder inequality and John-Nirenberg inequality,we demonstrate that the solutions are continuous with respect to parameter θ.Moreover,the necessary and sufficient conditions for robust optimal control are established using the linearization method.
基金The first author was supported by Hungarian National Research Fund OTKA No.K67819the second author was partially supported by Hungarian National Research Fund OTKA No.K67819the first and the third authors were supported by Jedlik project “ReCoMend”2008-2011。
文摘In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.
基金supported by the National Basic Research Program of China (973 Program) under Grant No.2007CB814904the National Natural Science Foundations of China under Grant Nos.10921101 and 10701050the Natural Science Foundation of Shandong Province under Grant Nos.JQ200801 and 2008BS01024
文摘Both necessary and sufficient maximum principles for optimal control of stochastic systemwith random jumps consisting of forward and backward state variables are proved.The control variableis allowed to enter both diffusion and jump coefficients.The result is applied to a mean-varianceportfolio selection mixed with a recursive utility functional optimization problem.Explicit expressionof the optimal portfolio selection strategy is obtained in the state feedback form.
文摘It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.
基金supported by Basic Research Program of China (Grant No.2007CB814904)National Natural Science Foundation of China (Grant No.10325101)Natural Science Foundation of Zhejiang Province (Grant No.Y605478,Y606667)
文摘The paper is concerned with a stochastic optimal control problem in which the controlled system is described by a fully coupled nonlinear forward-backward stochastic differential equation driven by a Brownian motion.It is required that all admissible control processes are adapted to a given subfiltration of the filtration generated by the underlying Brownian motion.For this type of partial information control,one sufficient(a verification theorem) and one necessary conditions of optimality are proved.The control domain need to be convex and the forward diffusion coefficient of the system can contain the control variable.
基金supported by the Natural Science Foundation of China under Grant No.61573217,111 Project(B12023)the National High-Level Personnel of Special Support Programthe Chang Jiang Scholar Program of Chinese Education Ministry
文摘This paper is concerned with the optimal control problems of forward-backward delay systems involving impulse controls. The authors establish a stochastic maximum principle for this kind of systems. The most distinguishing features of the proposed problem are that the control variables consist of regular and impulsive controls, both with time delay, and that the domain of regular control is not necessarily convex. The authors obtain the necessary and sufficient conditions for optimal controls,which have potential applications in mathematical finance.
基金This research is supported by the National Nature Science Foundation of China under Grant Nos 11001156, 11071144, the Nature Science Foundation of Shandong Province （ZR2009AQ017）, and Independent Innovation Foundation of Shandong University （IIFSDU）, China.
文摘This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backward stochastic differential equation theory with certain classical convex variational techniques, the necessary maximum principle is proved for the partially observed optimal control, where the control domain is a nonempty convex set. Under certain convexity assumptions, the author also gives the sufficient conditions of an optimal control for the aforementioned optimal optimal problem. To illustrate the theoretical result, the author also works out an example of partial information linear-quadratic optimal control, and finds an explicit expression of the corresponding optimal control by applying the necessary and sufficient maximum principle.
基金Research supported by NSF(No.11671231,11201262 and 10921101)Shandong Province(No.BS2013SF020 and ZR2014AP005)Young Scholars Program of Shandong University and the 111 Project(No.B12023).
文摘In this paper,we study the recursive stochastic optimal control problems.The control domain does not need to be convex,and the generator of the backward stochastic differential equation can contain z.We obtain the variational equations for backward stochastic differential equations,and then obtain the maximum principle which solves completely Peng’s open problem.
