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Akihiko Takahashi - One of the best experts on this subject based on the ideXlab platform.
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stochastic Differential Game in high frequency market
Automatica, 2019Co-Authors: Taiga Saito, Akihiko TakahashiAbstract:Abstract This paper presents an application of a linear quadratic stochastic Differential Game to a model in finance, which describes trading behaviors of different types of players in a high frequency stock market. Stability of the high frequency market is a central issue for financial markets. Building a model that expresses the trading behaviors of the different types of players and the price actions in turmoil is important to set regulations to maintain fair markets. Firstly, we represent trading behaviors of the three types of players, algorithmic traders, general traders, and market makers as well as the mid price process of a risky asset by a linear quadratic stochastic Differential Game. Secondly, we obtain a Nash equilibrium by solving a forward–backward stochastic Differential equation (FBSDE) derived from the stochastic maximum principle. Finally, we present numerical examples of the Nash equilibrium and the corresponding price action of the risky asset, which agree with the empirical findings on trading behaviors of players in high frequency markets. This model can be used to investigate the impact of regulation changes on the market stability as well as trading strategies of the players.
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stochastic Differential Game in high frequency market
Social Science Research Network, 2018Co-Authors: Taiga Saito, Akihiko TakahashiAbstract:This paper presents an application of a linear quadratic stochastic Differential Game to a financial model that describes trading behaviors of different types of players in a high frequency stock market. Stability of the stock market in a high frequency environment is a central issue in financial markets. Building a model that expresses the trading behaviors of the different types of players and the price actions in a high frequency market helps introduce better regulations for a healthy market. Firstly, we represent trading behaviors of the three types of players, algorithmic traders, general traders, and market makers as well as the mid-price process of a risky asset by a linear quadratic stochastic Differential Game. Secondly, we obtain a Nash equilibrium for open loop admissible strategies by solving a forward-backward stochastic Differential equation (FBSDE) derived from the stochastic maximum principle. Finally, we present numerical examples of the Nash equilibrium for open loop admissible strategies and the corresponding price action of the risky asset, which agree with the empirical findings on the mechanism of high frequency markets. This model can be used to investigate the impact of regulation changes on the market stability as well as trading strategies of the players.
Guangchen Wang - One of the best experts on this subject based on the ideXlab platform.
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linear quadratic stochastic stackelberg Differential Game with asymmetric information
Science in China Series F: Information Sciences, 2017Co-Authors: Jingtao Shi, Guangchen Wang, Jie XiongAbstract:This paper is concerned with a linear-quadratic stochastic Stackelberg Differential Game, where players have asymmetric roles, with one leader and one follower in the context of two-person Game. It is required that the information available to the follower is a sub-$\sigma$-algebra of the one of the leader. By maximum principle and optimal filtering, a feedback Stackelberg equilibrium of the Game is given. A special example is used to elaborate the result.
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leader follower stochastic Differential Game with asymmetric information and applications
arXiv: Optimization and Control, 2015Co-Authors: Jingtao Shi, Guangchen Wang, Jie XiongAbstract:This paper is concerned with a leader-follower stochastic Differential Game with asymmetric information, where the information available to the follower is based on some sub-$\sigma$-algebra of that available to the leader. Such kind of Game problem has wide applications in finance, economics and management engineering such as newsvendor problems, cooperative advertising and pricing problems. Stochastic maximum principles and verification theorems with partial information are obtained, to represent the Stackelberg equilibrium. As applications, a linear-quadratic leader-follower stochastic Differential Game with asymmetric information is studied. It is shown that the open-loop Stackelberg equilibrium admits a state feedback representation if some system of Riccati equations is solvable.
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a partial information non zero sum Differential Game of backward stochastic Differential equations with applications
Automatica, 2012Co-Authors: Guangchen WangAbstract:This paper is concerned with a new kind of non-zero sum Differential Game of backward stochastic Differential equations (BSDEs). It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motion. We establish a necessary condition in the form of maximum principle with Pontryagin's type for open-loop Nash equilibrium point of this type of partial information Game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) Game and a partial information financial problem.
David W. K. Yeung - One of the best experts on this subject based on the ideXlab platform.
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a cooperative stochastic Differential Game of transboundary industrial pollution
Automatica, 2008Co-Authors: David W. K. Yeung, Leon A. PetrosyanAbstract:Though cooperation in environmental control holds out the best promise of effective actions, limited success has been observed because existing multinational joint initiatives fail to satisfy the property of subGame consistency. A cooperative solution is subGame consistent if the solution optimality principle is maintained in any subGame which starts at a later time with any feasible state brought about by prior optimal behaviors. This paper presents a cooperative stochastic Differential Game of transboundary industrial pollution with two novel features. The first feature is that industrial production creates short-term local impacts and long-term global impacts on the environment. Secondly, a subGame consistent cooperative solution is derived in this stochastic Differential Game together with a payment distribution mechanism that supports the subGame consistent solution. This is the first time that pollution management is analyzed in a cooperative stochastic Differential Game framework under these novel features.
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dynamically consistent cooperative solution in a Differential Game of transboundary industrial pollution
Journal of Optimization Theory and Applications, 2007Co-Authors: David W. K. YeungAbstract:This paper presents a cooperative Differential Game of transboundary industrial pollution. A noted feature of the Game model is that the industrial sectors remain competitive among themselves while the governments cooperate in pollution abatement. It is the first time that time consistent solutions are derived in a cooperative Differential Game on pollution control with industries and governments being separate entities. A stochastic version of the model is presented and a subGame-consistent cooperative solution is provided. This is the first study of pollution management in a stochastic Differential Game framework.
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stochastic Differential Game model of a common property fishery
Journal of Optimization Theory and Applications, 1996Co-Authors: Steffen Jorgensen, David W. K. YeungAbstract:The paper presents a stochastic Differential Game model of a common-property commercial fishery and determines a feedback Nash equilibrium of the Game. Closed-form expressions for the value functions, the equilibrium harvesting strategies, and stationary distributions of the fish stock are derived. Sensitivity analyses with respect tot he model parameters are carried out. The paper also considers equilibrium outcomes under joint maximization and surplus maximization. In the latter case, an optimal market size (i.e., number of firms) is identified.
Valery Y Glizer - One of the best experts on this subject based on the ideXlab platform.
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Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game
'MDPI AG', 2021Co-Authors: Valery Y GlizerAbstract:A finite-horizon two-person non-zero-sum Differential Game is considered. The dynamics of the Game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the Game under the consideration is singular. A novel definition of the Nash equilibrium in this Game (a Nash equilibrium sequence) is proposed. The Game is solved by application of the regularization method. This method yields a new Differential Game, which is a regular Nash equilibrium Game. Moreover, the new Game is a partial cheap control Game. An asymptotic analysis of this Game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular Game is constructed. The expressions for the optimal values of the functionals in the singular Game are obtained. Illustrative examples are presented
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robust trajectory tracking Differential Game cheap control approach
International Journal of Systems Science, 2014Co-Authors: Vladimir Turetsky, Valery Y Glizer, Josef ShinarAbstract:A robust trajectory tracking problem is treated in the framework of a zero-sum linear-quadratic Differential Game of a general type. For the cheap control version of this Game, a novel solvability condition is derived. The sufficient condition, guaranteeing that the tracking problem is solved by the optimal strategy of the minimiser in the cheap control Game, is established. The boundedness of the time realisations of this strategy is analysed. An illustrative example is presented.
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complete solution of a Differential Game with linear dynamics and bounded controls
Applied Mathematics Research Express, 2010Co-Authors: Valery Y Glizer, Vladimir TuretskyAbstract:A Differential Game is an appropriate mathematical model for real-life control problems, which either involve many decision-makers or contain a high degree of uncertainties. There is a rich literature devoted to the theory of Differential Games (see e.g. [1–5]). A zero-sum finite-horizon Differential Game with linear dynamics and bounded controls was studied extensively in the literature, because of its considerable meaning both in theory and applications (see e.g. [6–10] and the references therein). Important applications of this Game are: a pursuit-evasion problem (see e.g. [11–14]), an airplane landing problem under windshear conditions (see [15] and references therein), and some others. Different versions of this Game were analyzed in the literature. A simple example with the ideal dynamics of the players was considered in [6]. The Game with a first-order
Meir Pachter - One of the best experts on this subject based on the ideXlab platform.
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Optimal Strategies of the Differential Game in a Circular Region
IEEE Control Systems Letters, 2020Co-Authors: Eloy Garcia, David W. Casbeer, Meir PachterAbstract:A two-player Differential Game is considered where a spy infiltrates the region of interest and a defender pursues and tries to capture the spy before it leaves such region. This letter provides the complete solution of this Game by characterizing the Barrier surface which separates the state space into the regions of win of the defender and the spy. In each region the optimal strategies that guarantee the prescribed outcome are provided. This is in contrast to recent results addressing the same problem where the proposed strategies do not guarantee capture of the spy by the defender when initially this outcome was prescribed by the Game of kind solution.
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toward a solution of the active target defense Differential Game
Dynamic Games and Applications, 2019Co-Authors: Meir Pachter, Eloy Garcia, David W. CasbeerAbstract:A novel pursuit-evasion Differential Game involving three agents is considered. An Attacker missile is pursuing a Target aircraft. The Target aircraft is aided by a Defender missile launched by, say, the wingman, to intercept the Attacker before it reaches the Target aircraft. Thus, a team is formed by the Target and the Defender which cooperate to maximize the separation between the Target aircraft and the point where the Attacker missile is intercepted by the Defender missile, while at the same time the Attacker tries to minimize said distance. A long-range Beyond Visual Range engagement which is in line with current CONcepts of OPeration is envisaged, and it is therefore assumed that the players have simple motion kinematics a la Isaacs. Also, the speed of the Attacker is equal to the speed of the Defender and the latter is interested in point capture. It is also assumed that at all time the Attacker is aware of the Defender’s position, i.e., it is a perfect information Game. The analytic/closed-form solution of the target defense pursuit-evasion Differential Game delineates the state space region where the Attacker can reach the Target without being intercepted by the Defender, thus disposing of the Game of Kind. The target defense Game of Degree is played in the remaining state space. The analytic solution of the Game of Degree yields the agents’ optimal state feedback strategies, that is, the instantaneous heading angles for the Target and the Defender team to maximize the terminal separation between Target and Attacker at the instant of interception of the Attacker by the Defender, and also the instantaneous optimal heading for the Attacker to minimize said separation. Their calculation hinges on the real-time solution of a quartic equation. In this paper we contribute to the solution of a Differential Game with three states—an additional example to the, admittedly small, repertoire of pursuit-evasion Differential Games in 3-D which can be solved in closed form.
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design and analysis of state feedback optimal strategies for the Differential Game of active defense
IEEE Transactions on Automatic Control, 2019Co-Authors: Eloy Garcia, David W. Casbeer, Meir PachterAbstract:This paper is concerned with a scenario of active target defense modeled as a zero-sum Differential Game. The Differential Game theory as developed by Isaacs provides the correct framework for the analysis of pursuit-evasion conflicts and the design of optimal strategies for the players involved in the Game. This paper considers an Attacker missile pursuing a Target aircraft protected by a Defender missile which aims at intercepting the Attacker before the latter reaches the Target aircraft. A Differential Game is formulated where the two opposing players/teams try to minimize/maximize the distance between the Target and the Attacker at the time of interception of the Attacker by the Defender and such time indicates the termination of the Game. The Attacker aims to minimize the terminal distance between itself and the Target at the moment of its interception by the Defender. The opposing player/team consists of two cooperating agents: The Target and the Defender. These two agents cooperate in order to accomplish the two objectives: Guarantee interception of the Attacker by the Defender and maximize the terminal Target-Attacker separation. In this paper, we provide a complete, closed form solution of the active target defense Differential Game; we synthesize closed-loop state feedback optimal strategies for the agents and obtain the Value function of the Game. We characterize the Target's escape set and show that the Value function is continuous and continuously differentiable over the Target's escape set, and that it satisfies the Hamilton–Jacobi–Isaacs equation everywhere in this set.
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a geometric approach for the cooperative two pursuer one evader Differential Game
IFAC-PapersOnLine, 2017Co-Authors: Eloy Garcia, David W. Casbeer, Zachariah E Fuchs, Dejan Milutinovic, Meir PachterAbstract:Abstract This paper revisits Isaacs’ two cutters and fugitive ship Differential Game where two faster pursuers cooperate to capture a slower evader in minimum time; the three players have simple motion. The evader, knowing that is being pursued by two cooperative pursuers, tries to maximize the capture time. A solution of this Differential Game is determined based on the Hamiltonian formulation and based on the geometric properties of the Game. Additionally, this solution is verified by analyzing the properties of the obtained candidate Value function.
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cooperative target defense Differential Game with a constrained maneuverable defender
Conference on Decision and Control, 2015Co-Authors: David W. Casbeer, Eloy Garcia, Zachariah E Fuchs, Meir PachterAbstract:This paper addresses the active target defense Differential Game where a Target aircraft is pursued by an Attacker missile, and a Defender missile is employed in order to intercept the Attacker and protect the evading Target. This paper extends the results concerning the active target defense Differential Game by allowing for the Defender's turning rate to be constrained. The restriction imposed on the Defender's turning rate is of great operational relevance since initially the Defender's heading might be different than the otherwise optimal heading when simple motion is assumed.
