Optimal impulse control and impulse games provide the cutting-edge frameworks for modeling systems where control actions occur at discrete time points,and optimizing objectives under discontinuous interventions.This r...Optimal impulse control and impulse games provide the cutting-edge frameworks for modeling systems where control actions occur at discrete time points,and optimizing objectives under discontinuous interventions.This review synthesizes the theoretical advancements,computational approaches,emerging challenges,and possible research directions in the field.Firstly,we briefly review the fundamental theory of continuous-time optimal control,including Pontryagin's maximum principle(PMP)and dynamic programming principle(DPP).Secondly,we present the foundational results in optimal impulse control,including necessary conditions and sufficient conditions.Thirdly,we systematize impulse game methodologies,from Nash equilibrium existence theory to the connection between Nash equilibrium and systems stability.Fourthly,we summarize the numerical algorithms including the intelligent computation approaches.Finally,we examine the new trends and challenges in theory and applications as well as computational considerations.展开更多
This paper investigates an international optimal investmentCconsumption problem under a random time horizon.The investor may allocate wealth between a domestic bond and an international real project with production ou...This paper investigates an international optimal investmentCconsumption problem under a random time horizon.The investor may allocate wealth between a domestic bond and an international real project with production output,whose price may exhibit discontinuities.The model incorporates the effects of taxation and exchange rate dynamics,where the exchange rate follows a stochastic differential equation with jump-diffusion.The investor’s objective is to maximize the utility of consumption and terminal wealth over an uncertain investment horizon.It is worth noting that,under our framework,the exit time is not assumed to be a stopping time.In particular,for the case of constant relative risk aversion(CRRA),we derive the optimal investment and consumption strategies by applying the separation method to solve the associated HamiltonCJacobiCBellman(HJB)equation.Moreover,several numerical examples are provided to illustrate the practical applicability of the proposed results.展开更多
This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and...This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control.In our game,both the(lower and upper)value functions and the(lower and upper)second-order Bellman–Isaacs equations are defined on the Wasserstein space P_(2)(R^(n))which is an infinite dimensional space.The dynamic programming principle for the value functions is proved.If the(upper and lower)value functions are smooth enough,we show that they are the classical solutions to the second-order Bellman–Isaacs equations.On the other hand,the classical solutions to the(upper and lower)Bellman–Isaacs equations are unique and coincide with the(upper and lower)value functions.As an illustrative application,the linear quadratic case is considered.Under the Isaacs condition,the explicit expressions of optimal closed-loop controls for both players are given.Finally,we introduce the intrinsic notion of viscosity solution of our second-order Bellman–Isaacs equations,and characterize the(upper and lower)value functions as their viscosity solutions.展开更多
We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, wher...We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.展开更多
This paper investigates an optimal investment strategy on consumption and portfolio problem, in which the investor must withdraw funds continuously at a given rate. By analyzing the evolving process of wealth, we give...This paper investigates an optimal investment strategy on consumption and portfolio problem, in which the investor must withdraw funds continuously at a given rate. By analyzing the evolving process of wealth, we give the definition of safe-region for investment. Moreover, in order to obtain the target wealth as quickly as possible, using Bellman dynamic programming principle, we get the optimal investment strategy and corresponding necessary expected time. At last we give some numerical computations for a set of different parameters.展开更多
We consider variations of the classical jeep problems： the optimal logistics for a caravan of jeeps which travel together in the desert. The main purpose is to arrange the travels for the one-way trip and the round t...We consider variations of the classical jeep problems： the optimal logistics for a caravan of jeeps which travel together in the desert. The main purpose is to arrange the travels for the one-way trip and the round trip of a caravan of jeeps so that the chief jeep visits the farthest destination. Based on the dynamic program principle, the maximum distances for the caravan when only part of the jeeps should return and when all drivers should return are obtained. Some related results such as the efficiency of the abandoned jeeps, and the advantages of more jeeps in the caravan are also presented.展开更多
The optimal bounded control of stochastic-excited systems with Duhem hysteretic components for maximizing system reliability is investigated. The Duhem hysteretic force is transformed to energy-depending damping and s...The optimal bounded control of stochastic-excited systems with Duhem hysteretic components for maximizing system reliability is investigated. The Duhem hysteretic force is transformed to energy-depending damping and stiffness by the energy dissipation balance technique. The controlled system is transformed to the equivalent non- hysteretic system. Stochastic averaging is then implemented to obtain the It5 stochastic equation associated with the total energy of the vibrating system, appropriate for eval- uating system responses. Dynamical programming equations for maximizing system re- liability are formulated by the dynamical programming principle. The optimal bounded control is derived from the maximization condition in the dynamical programming equation. Finally, the conditional reliability function and mean time of first-passage failure of the optimal Duhem systems are numerically solved from the Kolmogorov equations. The proposed procedure is illustrated with a representative example.展开更多
To enhance the reliability of the stochastically excited structure,it is significant to study the problem of stochastic optimal control for minimizing first-passage failure.Combining the stochastic averaging method wi...To enhance the reliability of the stochastically excited structure,it is significant to study the problem of stochastic optimal control for minimizing first-passage failure.Combining the stochastic averaging method with dynamical programming principle,we study the optimal control for minimizing first-passage failure of multidegrees-of-freedom(MDoF)nonlinear oscillators under Gaussian white noise excitations.The equations of motion of the controlled system are reduced to time homogenous difusion processes by stochastic averaging.The optimal control law is determined by the dynamical programming equations and the control constraint.The backward Kolmogorov(BK)equation and the Pontryagin equation are established to obtain the conditional reliability function and mean first-passage time(MFPT)of the optimally controlled system,respectively.An example has shown that the proposed control strategy can increase the reliability and MFPT of the original system,and the mathematical treatment is also facilitated.展开更多
A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is proposed. The nonlinear equation of cable motion in plane is derived and reduced to the equations ...A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is proposed. The nonlinear equation of cable motion in plane is derived and reduced to the equations for the first two modes of cable vibration by using the Galerkin method. The partially averaged Ito equation for controlled system energy is further derived by applying the stochastic averaging method for quasi-non-integrable Hamiltonian systems. The dynamical programming equation for the controlled system energy with a performance index is established by applying the stochastic dynamical programming principle and a stochastic optimal control law is obtained through solving the dynamical programming equation. A bilinear controller by using the direct method of Lyapunov is introduced. The comparison between the two controllers shows that the proposed stochastic optimal control strategy is superior to the bilinear control strategy in terms of higher control effectiveness and efficiency.展开更多
We study a new class of two-player,zero-sum,deterministic differential games where each player uses both continuous and impulse controls in an infinite horizon with discounted payoff.We assume that the form and cost o...We study a new class of two-player,zero-sum,deterministic differential games where each player uses both continuous and impulse controls in an infinite horizon with discounted payoff.We assume that the form and cost of impulses depend on nonlinear functions and the state of the system,respectively.We use Bellman's dynamic programming principle(DPP)and viscosity solutions approach to show,for this class of games,the existence and uniqueness of a solution for the associated Hamilton-Jacobi-Bellman-Isaacs(HJBI)partial differential equations(PDEs).We then,under Isaacs'condition,deduce that the lower and upper value functions coincide,and we give a computational procedure with a numerical test for the game.展开更多
This paper focuses on the McKean-Vlasov system's stochastic optimal control problem with Markov regime-switching.To this end,the authors establish a new It?'s formula using the linear derivative on the Wassers...This paper focuses on the McKean-Vlasov system's stochastic optimal control problem with Markov regime-switching.To this end,the authors establish a new It?'s formula using the linear derivative on the Wasserstein space.This formula enables us to derive the Hamilton-Jacobi-Bellman equation and verification theorems for Mc Kean-Vlasov optimal controls with regime-switching using dynamic programming.As concrete applications,the authors first study the McKean-Vlasov stochastic linear quadratic optimal control problem of the Markov regime-switching system,where all the coefficients can depend on the jump that switches among a finite number of states.Then,the authors represent the optimal control by four highly coupled Riccati equations.Besides,the authors revisit a continuous-time Markowitz mean-variance portfolio selection model(incomplete market)for a market consisting of one bank account and multiple stocks,in which the bank interest rate,the appreciation and volatility rates of the stocks are Markov-modulated.The mean-variance efficient portfolios can be derived explicitly in closed forms based on solutions of four Riccati equations.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
This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessa...This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessarily convex.Relations among the adjoint processes,the generalized Hamiltonian function and the value function are proven,under the assumption of a smooth value function and within the framework of viscosity solutions,respectively.Some examples are given to illustrate the theoretical results.展开更多
We will study the following problem.Let X_t,t∈[0,T],be an R^d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the informat...We will study the following problem.Let X_t,t∈[0,T],be an R^d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σ containing an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.展开更多
This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled...This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle （DPP） for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman （HJB） integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.展开更多
This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE ...This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE methods,in particular that of the notion from Peng’s stochastic backward semigroups,the authors prove a dynamic programming principle for both the upper and the lower value functions of the game.The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle.As a consequence,the uniqueness implies that the upper and lower value functions coincide and the game admits a value.展开更多
In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the d...In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.展开更多
文摘Optimal impulse control and impulse games provide the cutting-edge frameworks for modeling systems where control actions occur at discrete time points,and optimizing objectives under discontinuous interventions.This review synthesizes the theoretical advancements,computational approaches,emerging challenges,and possible research directions in the field.Firstly,we briefly review the fundamental theory of continuous-time optimal control,including Pontryagin's maximum principle(PMP)and dynamic programming principle(DPP).Secondly,we present the foundational results in optimal impulse control,including necessary conditions and sufficient conditions.Thirdly,we systematize impulse game methodologies,from Nash equilibrium existence theory to the connection between Nash equilibrium and systems stability.Fourthly,we summarize the numerical algorithms including the intelligent computation approaches.Finally,we examine the new trends and challenges in theory and applications as well as computational considerations.
基金Supported by the Shandong Provincial Natural Science Foundation(ZR2024MA095)Natural Science Foun-dation of China(12401583)Basic Research Program of Jiangsu(BK20240416).
文摘This paper investigates an international optimal investmentCconsumption problem under a random time horizon.The investor may allocate wealth between a domestic bond and an international real project with production output,whose price may exhibit discontinuities.The model incorporates the effects of taxation and exchange rate dynamics,where the exchange rate follows a stochastic differential equation with jump-diffusion.The investor’s objective is to maximize the utility of consumption and terminal wealth over an uncertain investment horizon.It is worth noting that,under our framework,the exit time is not assumed to be a stopping time.In particular,for the case of constant relative risk aversion(CRRA),we derive the optimal investment and consumption strategies by applying the separation method to solve the associated HamiltonCJacobiCBellman(HJB)equation.Moreover,several numerical examples are provided to illustrate the practical applicability of the proposed results.
基金supported by Natural Science Foundation of Shandong Province(Grant Nos.ZR2020MA032,ZR2022MA029)National Natural Science Foundation of China(Grant Nos.12171279,72171133)+1 种基金The second named author was supported by National Key R&D Program of China(Grant No.2022YFA1006102)National Natural Science Foundation of China(Grant No.11831010)。
文摘This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control.In our game,both the(lower and upper)value functions and the(lower and upper)second-order Bellman–Isaacs equations are defined on the Wasserstein space P_(2)(R^(n))which is an infinite dimensional space.The dynamic programming principle for the value functions is proved.If the(upper and lower)value functions are smooth enough,we show that they are the classical solutions to the second-order Bellman–Isaacs equations.On the other hand,the classical solutions to the(upper and lower)Bellman–Isaacs equations are unique and coincide with the(upper and lower)value functions.As an illustrative application,the linear quadratic case is considered.Under the Isaacs condition,the explicit expressions of optimal closed-loop controls for both players are given.Finally,we introduce the intrinsic notion of viscosity solution of our second-order Bellman–Isaacs equations,and characterize the(upper and lower)value functions as their viscosity solutions.
基金supported by the NSF of China(11071144,11171187,11222110 and 71671104)Shandong Province(BS2011SF010,JQ201202)+4 种基金SRF for ROCS(SEM)Program for New Century Excellent Talents in University(NCET-12-0331)111 Project(B12023)the Ministry of Education of Humanities and Social Science Project(16YJA910003)Incubation Group Project of Financial Statistics and Risk Management of SDUFE
文摘We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.
文摘This paper investigates an optimal investment strategy on consumption and portfolio problem, in which the investor must withdraw funds continuously at a given rate. By analyzing the evolving process of wealth, we give the definition of safe-region for investment. Moreover, in order to obtain the target wealth as quickly as possible, using Bellman dynamic programming principle, we get the optimal investment strategy and corresponding necessary expected time. At last we give some numerical computations for a set of different parameters.
基金partially Supported by National Natural Science Foundation of China(70571079,60534080)China Postdoctoral Science Foundation(20100471140)
文摘We consider variations of the classical jeep problems： the optimal logistics for a caravan of jeeps which travel together in the desert. The main purpose is to arrange the travels for the one-way trip and the round trip of a caravan of jeeps so that the chief jeep visits the farthest destination. Based on the dynamic program principle, the maximum distances for the caravan when only part of the jeeps should return and when all drivers should return are obtained. Some related results such as the efficiency of the abandoned jeeps, and the advantages of more jeeps in the caravan are also presented.
基金supported by the National Natural Science Foundation of China(Nos.11202181 and11402258)the Special Fund for the Doctoral Program of Higher Education of China(No.20120101120171)
文摘The optimal bounded control of stochastic-excited systems with Duhem hysteretic components for maximizing system reliability is investigated. The Duhem hysteretic force is transformed to energy-depending damping and stiffness by the energy dissipation balance technique. The controlled system is transformed to the equivalent non- hysteretic system. Stochastic averaging is then implemented to obtain the It5 stochastic equation associated with the total energy of the vibrating system, appropriate for eval- uating system responses. Dynamical programming equations for maximizing system re- liability are formulated by the dynamical programming principle. The optimal bounded control is derived from the maximization condition in the dynamical programming equation. Finally, the conditional reliability function and mean time of first-passage failure of the optimal Duhem systems are numerically solved from the Kolmogorov equations. The proposed procedure is illustrated with a representative example.
基金the National Natural Science Foundation of China(Nos.11272201,11132007 and 10802030)
文摘To enhance the reliability of the stochastically excited structure,it is significant to study the problem of stochastic optimal control for minimizing first-passage failure.Combining the stochastic averaging method with dynamical programming principle,we study the optimal control for minimizing first-passage failure of multidegrees-of-freedom(MDoF)nonlinear oscillators under Gaussian white noise excitations.The equations of motion of the controlled system are reduced to time homogenous difusion processes by stochastic averaging.The optimal control law is determined by the dynamical programming equations and the control constraint.The backward Kolmogorov(BK)equation and the Pontryagin equation are established to obtain the conditional reliability function and mean first-passage time(MFPT)of the optimally controlled system,respectively.An example has shown that the proposed control strategy can increase the reliability and MFPT of the original system,and the mathematical treatment is also facilitated.
基金supported by the National Natural Science Foundation of China (11072212,10932009)the Zhejiang Natural Science Foundation of China (7080070)
文摘A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is proposed. The nonlinear equation of cable motion in plane is derived and reduced to the equations for the first two modes of cable vibration by using the Galerkin method. The partially averaged Ito equation for controlled system energy is further derived by applying the stochastic averaging method for quasi-non-integrable Hamiltonian systems. The dynamical programming equation for the controlled system energy with a performance index is established by applying the stochastic dynamical programming principle and a stochastic optimal control law is obtained through solving the dynamical programming equation. A bilinear controller by using the direct method of Lyapunov is introduced. The comparison between the two controllers shows that the proposed stochastic optimal control strategy is superior to the bilinear control strategy in terms of higher control effectiveness and efficiency.
基金supported by the National Center for Scientific and Technical Research CNRST,Rabat,Morocco[grant number 17 UIZ 19].
文摘We study a new class of two-player,zero-sum,deterministic differential games where each player uses both continuous and impulse controls in an infinite horizon with discounted payoff.We assume that the form and cost of impulses depend on nonlinear functions and the state of the system,respectively.We use Bellman's dynamic programming principle(DPP)and viscosity solutions approach to show,for this class of games,the existence and uniqueness of a solution for the associated Hamilton-Jacobi-Bellman-Isaacs(HJBI)partial differential equations(PDEs).We then,under Isaacs'condition,deduce that the lower and upper value functions coincide,and we give a computational procedure with a numerical test for the game.
基金supported partly by the National Nature Science Foundation of China under Grant Nos.12171053,11701040,11871010,61871058the Fundamental Research Funds for the Central Universities of Renmin University of China under Grant No.23XNKJ05the Hong Kong General Research Fund under Grant Nos.15216720,15221621 and 15226922。
文摘This paper focuses on the McKean-Vlasov system's stochastic optimal control problem with Markov regime-switching.To this end,the authors establish a new It?'s formula using the linear derivative on the Wasserstein space.This formula enables us to derive the Hamilton-Jacobi-Bellman equation and verification theorems for Mc Kean-Vlasov optimal controls with regime-switching using dynamic programming.As concrete applications,the authors first study the McKean-Vlasov stochastic linear quadratic optimal control problem of the Markov regime-switching system,where all the coefficients can depend on the jump that switches among a finite number of states.Then,the authors represent the optimal control by four highly coupled Riccati equations.Besides,the authors revisit a continuous-time Markowitz mean-variance portfolio selection model(incomplete market)for a market consisting of one bank account and multiple stocks,in which the bank interest rate,the appreciation and volatility rates of the stocks are Markov-modulated.The mean-variance efficient portfolios can be derived explicitly in closed forms based on solutions of four Riccati equations.
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supported by National Key Research and Development Program of China under Grant No.2022YFA1006104the National Natural Science Foundations of China under Grant Nos.12471419 and 12271304the Natural Science Foundation of Shandong Province under Grant No.ZR2022JQ01。
文摘This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessarily convex.Relations among the adjoint processes,the generalized Hamiltonian function and the value function are proven,under the assumption of a smooth value function and within the framework of viscosity solutions,respectively.Some examples are given to illustrate the theoretical results.
基金Supported in part by National Natural Science Foundation of China Grant (No.10131040).The author also thanks the referee's constructive suggestions.
文摘We will study the following problem.Let X_t,t∈[0,T],be an R^d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σ containing an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.
基金supported by the National Natural Science Foundation of China under Grant Nos.11171187,11222110Shandong Province under Grant No.JQ201202+1 种基金Program for New Century Excellent Talents in University under Grant No.NCET-12-0331111 Project under Grant No.B12023
文摘This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle （DPP） for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman （HJB） integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.
基金supported by the National Nature Science Foundation of China under Grant Nos.11701040,11871010,61871058the Fundamental Research Funds for the Central Universities under Grant No.2019XDA11。
文摘This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE methods,in particular that of the notion from Peng’s stochastic backward semigroups,the authors prove a dynamic programming principle for both the upper and the lower value functions of the game.The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle.As a consequence,the uniqueness implies that the upper and lower value functions coincide and the game admits a value.
基金supported by the Agence Nationale de la Recherche (France), reference ANR-10-BLAN 0112the Marie Curie ITN "Controlled Systems", call: FP7-PEOPLE-2007-1-1-ITN, no. 213841-2+3 种基金supported by the National Natural Science Foundation of China (No. 10701050, 11071144)National Basic Research Program of China (973 Program) (No. 2007CB814904)Shandong Province (No. Q2007A04),Independent Innovation Foundation of Shandong Universitythe Project-sponsored by SRF for ROCS, SEM
文摘In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.
