In this paper,we study the n-player game and the mean field game under the constant relative risk aversion relative performance on terminal wealth,in which the interaction occurs by peer competition.In the model with ...In this paper,we study the n-player game and the mean field game under the constant relative risk aversion relative performance on terminal wealth,in which the interaction occurs by peer competition.In the model with n agents,the price dynamics of underlying risky assets depend on a common noise and contagious jump risk modeled by a multi-dimensional nonlinear Hawkes process.With a continuum of agents,we formulate the mean field game problem and characterize a deterministic mean field equilibrium in an analytical form under some conditions,allowing us to investigate some impacts of model parameters in the limiting model and discuss some financial implications.Moreover,based on the mean field equilibrium,we construct an approximate Nash equilibrium for the n-player game when n is sufficiently large.The explicit order of the approximation error is also derived.展开更多
We study n-player games of portfolio choice in general common Ito-diffusion markets under relative performance criteria and time monotone forward utilities.We,also,consider their continuum limit which gives rise to a ...We study n-player games of portfolio choice in general common Ito-diffusion markets under relative performance criteria and time monotone forward utilities.We,also,consider their continuum limit which gives rise to a forward mean field game with unbounded controls in both the drift and volatility terms.Furthermore,we allow for general(time monotone)preferences,thus departing from the homothetic case,the only case so far analyzed.We produce explicit solutions for the optimal policies,the optimal wealth processes and the game values,and also provide representative examples for both the finite and the mean field game.展开更多
In this paper,we focus on mean-field linear-quadratic games for stochastic large-population systems with time delays.The e-Nash equilibrium for decentralized strategies in linear-quadratic games is derived via the con...In this paper,we focus on mean-field linear-quadratic games for stochastic large-population systems with time delays.The e-Nash equilibrium for decentralized strategies in linear-quadratic games is derived via the consistency condition.By means of variational analysis,the system of consistency conditions can be expressed by forward-backward stochastic differential equations.Numerical examples illustrate the sensitivity of solutions of advanced backward stochastic differential equations to time delays,the effect of the the population's collective behaviors,and the consistency of mean-field estimates.展开更多
This paper designs an incentive Stackelberg strategy for the discrete-time stochastic systems with mean-field terms.Sufficient conditions for the existence of such a design are suggested.Moreover,the incentive strateg...This paper designs an incentive Stackelberg strategy for the discrete-time stochastic systems with mean-field terms.Sufficient conditions for the existence of such a design are suggested.Moreover,the incentive strategy is obtained as a feedback form including the deviation of the state and its mathematical expectation.Also,the stability analysis is involved.It is found that the stability can be guaranteed by the follower.In addition,the specific algorithm is proposed and its effectiveness is checked by two examples.展开更多
This paper considers a linear-quadratic(LQ) meanfield game governed by a forward-backward stochastic system with partial observation and common noise,where a coupling structure enters state equations,cost functionals ...This paper considers a linear-quadratic(LQ) meanfield game governed by a forward-backward stochastic system with partial observation and common noise,where a coupling structure enters state equations,cost functionals and observation equations.Firstly,to reduce the complexity of solving the meanfield game,a limiting control problem is introduced.By virtue of the decomposition approach,an admissible control set is proposed.Applying a filter technique and dimensional-expansion technique,a decentralized control strategy and a consistency condition system are derived,and the related solvability is also addressed.Secondly,we discuss an approximate Nash equilibrium property of the decentralized control strategy.Finally,we work out a financial problem with some numerical simulations.展开更多
Unmanned Aerial Vehicles(UAVs)enabled Aerial Base Stations(UABSs)have been studied widely in future communications.However,there are a series of challenges such as interference management,trajectory design and resourc...Unmanned Aerial Vehicles(UAVs)enabled Aerial Base Stations(UABSs)have been studied widely in future communications.However,there are a series of challenges such as interference management,trajectory design and resource allocation in the scenarios of multi-UAV networks.Besides,different performances among UABSs increase complexity and bring many challenges.In this paper,the joint downlink transmission power control and trajectory design problem in multi-type UABSs communication network is investigated.In order to satisfy the signal to interference plus noise power ratio of users,each UABS needs to adjust its position and transmission power.Based on the interactions among multiple communication links,a non-cooperative Mean-Field-Type Game(MFTG)is proposed to model the joint optimization problem.Then,a Nash equilibrium solution is solved by two steps:first,the users in the given area are clustered to get the initial deployment of the UABSs;second,the Mean-Field Q(MFQ)-learning algorithm is proposed to solve the discrete MFTG problem.Finally,the effectiveness of the approach is verified through the simulations,which simplifies the solution process and effectively reduces the energy consumption of each UABS.展开更多
With the development of the Internet of Things,the edge devices are increasing.Cyber security issues in edge computing have also emerged and caused great concern.We propose a defense strategy based on Mean field game ...With the development of the Internet of Things,the edge devices are increasing.Cyber security issues in edge computing have also emerged and caused great concern.We propose a defense strategy based on Mean field game to solve the security issues of edge user data during edge computing.Firstly,an individual cost function is formulated to build an edge user data security defense model.Secondly,we research the𝜀𝜀-Nash equilibrium of the individual cost function with finite players and prove the existence of the optimal defense strategy.Finally,by analyzing the stability of edge user data loss,it proves that the proposed defense strategy is effective.展开更多
为了简化使用完美马尔科夫均衡方法可能引起的复杂计算过程,本文依据博弈论方法,提出一种平均场均衡的无线自组织网络路由协议(mean field equilibrium AODV,MFEA)。该方法要求每个节点利用所有其他节点的信息来分析自己的最优策略,而...为了简化使用完美马尔科夫均衡方法可能引起的复杂计算过程,本文依据博弈论方法,提出一种平均场均衡的无线自组织网络路由协议(mean field equilibrium AODV,MFEA)。该方法要求每个节点利用所有其他节点的信息来分析自己的最优策略,而不需要知道每一个局中人的信息,并且在足够大的局中人数目情况下性能更加近似马尔科夫均衡。仿真实验显示:提出的MFEA路由协议在包投递率、时延和归一化开销方面均优于AODV(Ad hoc on-demand distance vector routing)协议,在节点密集的无线自组织网络中仍可获得比较好效果。展开更多
With energy harvesting capability, the Internet of things(IoT) devices transmit data depending on their available energy, which leads to a more complicated coupling and brings new technical challenges to delay optimiz...With energy harvesting capability, the Internet of things(IoT) devices transmit data depending on their available energy, which leads to a more complicated coupling and brings new technical challenges to delay optimization. In this paper,we study the delay-optimal random access(RA) in large-scale energy harvesting IoT networks. We model a two-dimensional Markov decision process(MDP)to address the coupling between the data and energy queues, and adopt the mean field game(MFG) theory to reveal the coupling among the devices by utilizing the large-scale property. Specifically, to obtain the optimal access strategy for each device, we derive the Hamilton-Jacobi-Bellman(HJB) equation which requires the statistical information of other devices.Moreover, to model the evolution of the states distribution in the system, we derive the Fokker-PlanckKolmogorov(FPK) equation based on the access strategy of devices. By solving the two coupled equations,we obtain the delay-optimal random access solution in an iterative manner with Lax-Friedrichs method. Finally, the simulation results show that the proposed scheme achieves significant performance gain compared with the conventional schemes.展开更多
We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “cons...We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website:?http://www.econ.univpm.it/recchioni/finance.展开更多
The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control.It is well known that,for standard mean field games,cert...The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control.It is well known that,for standard mean field games,certain monotonicity conditions are crucial to guarantee the uniqueness of mean field equilibria and then the global wellposedness for master equations.In the literature the monotonicity condition could be the Lasry–Lions monotonicity,the displacement monotonicity,or the anti-monotonicity conditions.In this paper,we investigate these three types of monotonicity conditions for Mean Field Games of Controls and show their propagation along the solutions to the master equations with common noises.In particular,we extend the displacement monotonicity to semi-monotonicity,whose propagation result is new even for standard mean field games.This is the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls.展开更多
This paper studies the backward-forward linear-quadratic-Gaussian(LQG)games with major and minor agents(players).The state of major agent follows a linear backward stochastic differential equation(BSDE)and the states ...This paper studies the backward-forward linear-quadratic-Gaussian(LQG)games with major and minor agents(players).The state of major agent follows a linear backward stochastic differential equation(BSDE)and the states of minor agents are governed by linear forward stochastic differential equations(SDEs).The major agent is dominating as its state enters those of minor agents.On the other hand,all minor agents are individually negligible but their state-average affects the cost functional of major agent.The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies.We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation(BFSDE)system.Next,we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method.Consequently,we obtain the decentralized strategies for major and minor agents which are proved to satisfy the-Nash equilibrium property.展开更多
基金supported by Natural Science Basic Research Program of Shaanxi(Grant No.2023-JC-JQ-05)National Natural Science Foundation of China(Grant No.11971368)+1 种基金supported by the Fundamental Research Funds for the Central Universities(Grant No.WK3470000024)supported by The Hong Kong Polytechnic University(Grant Nos.P0031417 and P0039251)。
文摘In this paper,we study the n-player game and the mean field game under the constant relative risk aversion relative performance on terminal wealth,in which the interaction occurs by peer competition.In the model with n agents,the price dynamics of underlying risky assets depend on a common noise and contagious jump risk modeled by a multi-dimensional nonlinear Hawkes process.With a continuum of agents,we formulate the mean field game problem and characterize a deterministic mean field equilibrium in an analytical form under some conditions,allowing us to investigate some impacts of model parameters in the limiting model and discuss some financial implications.Moreover,based on the mean field equilibrium,we construct an approximate Nash equilibrium for the n-player game when n is sufficiently large.The explicit order of the approximation error is also derived.
文摘We study n-player games of portfolio choice in general common Ito-diffusion markets under relative performance criteria and time monotone forward utilities.We,also,consider their continuum limit which gives rise to a forward mean field game with unbounded controls in both the drift and volatility terms.Furthermore,we allow for general(time monotone)preferences,thus departing from the homothetic case,the only case so far analyzed.We produce explicit solutions for the optimal policies,the optimal wealth processes and the game values,and also provide representative examples for both the finite and the mean field game.
基金supported by the National Natural Science Foundation of China (Grant No.11801154)R.J.Li was supported by the Guangzhou Science and Technology Program Project Project (Grant No.202201011057)W.F.Wang was supported by the Natural Science Foundation of Hubei Province (Grant No.2023AFC006).
文摘In this paper,we focus on mean-field linear-quadratic games for stochastic large-population systems with time delays.The e-Nash equilibrium for decentralized strategies in linear-quadratic games is derived via the consistency condition.By means of variational analysis,the system of consistency conditions can be expressed by forward-backward stochastic differential equations.Numerical examples illustrate the sensitivity of solutions of advanced backward stochastic differential equations to time delays,the effect of the the population's collective behaviors,and the consistency of mean-field estimates.
基金supported by the National Natural Science Foundation of China under Grant Nos.61903234 and 61973198the Natural Science Foundation of Shandong Province under Grant No.ZR2021MA066。
文摘This paper designs an incentive Stackelberg strategy for the discrete-time stochastic systems with mean-field terms.Sufficient conditions for the existence of such a design are suggested.Moreover,the incentive strategy is obtained as a feedback form including the deviation of the state and its mathematical expectation.Also,the stability analysis is involved.It is found that the stability can be guaranteed by the follower.In addition,the specific algorithm is proposed and its effectiveness is checked by two examples.
基金supported by the National Key Research and Development Program of China(2022YFA1006103,2023YFA1009203)the National Natural Science Foundation of China(61925306,61821004,11831010,61977043,12001320)+2 种基金the Natural Science Foundation of Shandong Province(ZR2019ZD42,ZR2020ZD24)the Taishan Scholars Young Program of Shandong(TSQN202211032)the Young Scholars Program of Shandong University。
文摘This paper considers a linear-quadratic(LQ) meanfield game governed by a forward-backward stochastic system with partial observation and common noise,where a coupling structure enters state equations,cost functionals and observation equations.Firstly,to reduce the complexity of solving the meanfield game,a limiting control problem is introduced.By virtue of the decomposition approach,an admissible control set is proposed.Applying a filter technique and dimensional-expansion technique,a decentralized control strategy and a consistency condition system are derived,and the related solvability is also addressed.Secondly,we discuss an approximate Nash equilibrium property of the decentralized control strategy.Finally,we work out a financial problem with some numerical simulations.
基金co-supported by the National Natural Science Foundation of China(Nos.62001387,61901379)the Natural Science Basic Research Plan in Shaanxi Province(No.2019JQ253)+4 种基金the Key R&D Plan of Shaanxi Province(No.2020GY034)the Aerospace Science and Technology Innovation Fund of China Aerospace Science and Technology Corporationthe Shanghai Aerospace Science and Technology Innovation Fund(No.SAST2018045)the China Fundamental Research Fund for the Central Universities(No.3102018QD096)the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University(No.CX2020152)。
文摘Unmanned Aerial Vehicles(UAVs)enabled Aerial Base Stations(UABSs)have been studied widely in future communications.However,there are a series of challenges such as interference management,trajectory design and resource allocation in the scenarios of multi-UAV networks.Besides,different performances among UABSs increase complexity and bring many challenges.In this paper,the joint downlink transmission power control and trajectory design problem in multi-type UABSs communication network is investigated.In order to satisfy the signal to interference plus noise power ratio of users,each UABS needs to adjust its position and transmission power.Based on the interactions among multiple communication links,a non-cooperative Mean-Field-Type Game(MFTG)is proposed to model the joint optimization problem.Then,a Nash equilibrium solution is solved by two steps:first,the users in the given area are clustered to get the initial deployment of the UABSs;second,the Mean-Field Q(MFQ)-learning algorithm is proposed to solve the discrete MFTG problem.Finally,the effectiveness of the approach is verified through the simulations,which simplifies the solution process and effectively reduces the energy consumption of each UABS.
文摘With the development of the Internet of Things,the edge devices are increasing.Cyber security issues in edge computing have also emerged and caused great concern.We propose a defense strategy based on Mean field game to solve the security issues of edge user data during edge computing.Firstly,an individual cost function is formulated to build an edge user data security defense model.Secondly,we research the𝜀𝜀-Nash equilibrium of the individual cost function with finite players and prove the existence of the optimal defense strategy.Finally,by analyzing the stability of edge user data loss,it proves that the proposed defense strategy is effective.
文摘为了简化使用完美马尔科夫均衡方法可能引起的复杂计算过程,本文依据博弈论方法,提出一种平均场均衡的无线自组织网络路由协议(mean field equilibrium AODV,MFEA)。该方法要求每个节点利用所有其他节点的信息来分析自己的最优策略,而不需要知道每一个局中人的信息,并且在足够大的局中人数目情况下性能更加近似马尔科夫均衡。仿真实验显示:提出的MFEA路由协议在包投递率、时延和归一化开销方面均优于AODV(Ad hoc on-demand distance vector routing)协议,在节点密集的无线自组织网络中仍可获得比较好效果。
基金supported in part by Key R&D Program of Zhejiang (No. 2022C03078)National Natural Science Foundation of China (No. U20A20158)+1 种基金National Key R&D Program of China (No. 2018YFB1801104)Ningbo S&T Major Project (No. 2019B10079)。
文摘With energy harvesting capability, the Internet of things(IoT) devices transmit data depending on their available energy, which leads to a more complicated coupling and brings new technical challenges to delay optimization. In this paper,we study the delay-optimal random access(RA) in large-scale energy harvesting IoT networks. We model a two-dimensional Markov decision process(MDP)to address the coupling between the data and energy queues, and adopt the mean field game(MFG) theory to reveal the coupling among the devices by utilizing the large-scale property. Specifically, to obtain the optimal access strategy for each device, we derive the Hamilton-Jacobi-Bellman(HJB) equation which requires the statistical information of other devices.Moreover, to model the evolution of the states distribution in the system, we derive the Fokker-PlanckKolmogorov(FPK) equation based on the access strategy of devices. By solving the two coupled equations,we obtain the delay-optimal random access solution in an iterative manner with Lax-Friedrichs method. Finally, the simulation results show that the proposed scheme achieves significant performance gain compared with the conventional schemes.
文摘We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website:?http://www.econ.univpm.it/recchioni/finance.
基金Chenchen Mou is supported in part by CityU Start-up(Grant No.7200684)Hong Kong RGC(Grant No.ECS 9048215).Jianfeng Zhang is supported in part by NSF(Grant Nos.DMS-1908665 and DMS-2205972).
文摘The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control.It is well known that,for standard mean field games,certain monotonicity conditions are crucial to guarantee the uniqueness of mean field equilibria and then the global wellposedness for master equations.In the literature the monotonicity condition could be the Lasry–Lions monotonicity,the displacement monotonicity,or the anti-monotonicity conditions.In this paper,we investigate these three types of monotonicity conditions for Mean Field Games of Controls and show their propagation along the solutions to the master equations with common noises.In particular,we extend the displacement monotonicity to semi-monotonicity,whose propagation result is new even for standard mean field games.This is the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls.
基金support partly by RGC Grant 502412,15300514,G-YL04.ZWu acknowledges the Natural Science Foundation of China(61573217),111 project(B12023)the National High-level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry.
文摘This paper studies the backward-forward linear-quadratic-Gaussian(LQG)games with major and minor agents(players).The state of major agent follows a linear backward stochastic differential equation(BSDE)and the states of minor agents are governed by linear forward stochastic differential equations(SDEs).The major agent is dominating as its state enters those of minor agents.On the other hand,all minor agents are individually negligible but their state-average affects the cost functional of major agent.The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies.We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation(BFSDE)system.Next,we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method.Consequently,we obtain the decentralized strategies for major and minor agents which are proved to satisfy the-Nash equilibrium property.
