Differential games with continuous,switching and impulse controls

A two-person zero-sum differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching, and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott–Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.     

Recurrence and Ergodicity of Switching Diffusions with Past-Dependent Switching Having a Countable State Space

This work focuses on recurrence and ergodicity of switching diffusions consisting of continuous and discrete components, in which the discrete component takes values in a countably infinite set and the rates of switching at current time depend on the value of the continuous component over an interval including certain past history. Sufficient conditions for recurrence and ergodicity are given. Moreover, the relationship between systems of partial differential equations and recurrence when the switching is past-independent is established under suitable conditions.     

Hilbert空间中具无界控制的无限时区线性二次最优控制问题

本文研究了Hilbert空间中一类由解析半群支配的具无界控制的无限时区线性二次最优控制问题,其中指标中的控制项加权算子要求强制而状态项加权算子可允许为不定号.在指数能稳条件下,证明了任意的最优控制及其最优轨线必定连续,建立了正实引理作为此问题唯一可解的充要条件,并用代数Riccati方程的解给出了最优控制的闭环综合。     

Zero-Sum Differential Games Involving Hybrid Controls

We study a zero-sum differential game with hybrid controls in which both players are allowed to use continuous as well as discrete controls. Discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, an autonomous jump set A or a controlled jump set C, where one controller can choose to jump or not. At each jump, the trajectory can move to a different Euclidean space. One player uses all the three types of controls, namely, continuous controls, autonomous jumps, and controlled jumps; the other player uses continuous controls and autonomous jumps. We prove the continuity of the associated lower and upper value functions V⁻ and V⁺. Using the dynamic programming principle satisfied by V⁻ and V⁺, we derive lower and upper quasivariational inequalities satisfied in the viscosity sense. We characterize the lower and upper value functions as the unique viscosity solutions of the corresponding quasivariational inequalities. Lastly, we state an Isaacs like condition for the game to have a value This work was partially supported by Grants DRDO 508 and ISRO 050 to the Non-linear Studies Group, Indian Institute of Science. The first author is a University Grant Commission Research Fellow and the financial support is gratefully acknowledged. The authors thank Prof. M.K. Ghosh, Department of Mathematics, Indian Institute of Science, for introducing the problem and thank the referee for useful suggestions.     

On the existence of Nash equilibria in large games

We study the existence of Nash equilibria in games with an infinite number of players. We show that there exists a Nash equilibrium in mixed strategies in all normal form games such that pure strategy sets are compact metric spaces and utility functions are continuous. The player set can be any nonempty set.     

The bayesian formulation of incomplete information — The non-compact case

In a game with incomplete information, a player may have beliefs about nature, about the other players' beliefs about nature, and so on, in an infinite hierarchy. We generalize a construction of Mertens & Zamir and show, that if nature is any Hausdorff space, and beliefs are regular Borel probability measures, then the space of all such infinite hierarchies of the players is a product of nature and the types of every player, where a type of a player is a belief about nature and the other players' types.     

Convex semi-infinite games

This paper introduces a generalization of semi-infinite games. The pure strategies for player I involve choosing one function from an infinite family of convex functions, while the set of mixed strategies for player II is a closed convex setC inR ⁿ. The minimax theorem applies under a condition which limits the directions of recession ofC. Player II always has optimal strategies. These are shown to exist for player I also if a certain infinite system verifies the property of Farkas-Minkowski. The paper also studies certain conditions that guarantee the finiteness of the value of the game and the existence of optimal pure strategies for player I.Many thanks are due to the referees for their detailed comments.     

The tug-of-war without noise and the infinity Laplacian in a wedge

Consider the ending time of the tug-of-war without noise in a wedge. There is a critical angle for finiteness of its expectation when player I maximizes the distance to the boundary and player II minimizes the distance. There is also a critical angle such that for smaller angles, player II can find a strategy where the expected ending time is finite, regardless of player I’s strategy. For larger angles, for each strategy of player II, player I can find a strategy making the expected ending time infinite. Using connections with the inhomogeneous infinity Laplacian, we bound this critical angle.     

Bayesian semiparametric Markov switching stochastic volatility model

This paper proposes a novel Bayesian semiparametric stochastic volatility model with Markov switching regimes for modeling the dynamics of the financial returns. The distribution of the error term of the returns is modeled as an infinite mixture of Normals; meanwhile, the intercept of the volatility equation is allowed to switch between two regimes. The proposed model is estimated using a novel sequential Monte Carlo method called particle learning that is especially well suited for state‐space models. The model is tested on simulated data and, using real financial times series, compared to a model without the Markov switching regimes. The results show that including a Markov switching specification provides higher predictive power for the entire distribution, as well as in the tails of the distribution. Finally, the estimate of the persistence parameter decreases significantly, a finding consistent with previous empirical studies.     

Convexity and characterization of optimal policies in a dynamic routing problem

An infinite horizon, expected average cost, dynamic routing problem is formulated for a simple failure-prone queueing system, modelled as a continuous time, continuous state controlled stochastic process. We prove that the optimal average cost is independent of the initial state and that the cost-to-go functions of dynamic programming are convex. These results, together with a set of optimality conditions, lead to the conclusion that optimal policies are switching policies, characterized by a set of switching curves (or regions), each curve corresponding to a particular state of the nodes (servers).     

Games of stopping with infinite horizon

We consider two-person zero-sum games of stopping: two players sequentially observe a stochastic process with infinite time horizon. Player I selects a stopping time and player II picks the distribution of the process. The pay-off is given by the expected value of the stopped process. Results of Irle (1990) on existence of value and equivalence of randomization for such games with finite time horizon, where the set of strategies for player II is dominated in the measure-theoretical sense, are extended to the infinite time case. Furthermore we treat such games when the set of strategies for player II is not dominated. A counterexample shows that even in the finite time case such games may not have a value. Then a sufficient condition for the existence of value is given which applies to prophet-type games.     

On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay

In this paper, we study the solvability and optimal controls of a class of fractional integrodifferential evolution systems with infinite delay in Banach spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and continuous dependence of mild solutions are investigated by utilizing the techniques of a priori estimation and extension of step by steps. Finally, existence of optimal controls for system governed by fractional integrodifferential evolution systems with infinite delay is proved.     

有限时区非线性系统的最优切换控制

This paper proposes a optimal control problem for a general nonlinear systems with finitely many admissible control settings and with costs assigned to switching of controls. With dynamic programming and viscosity solution theory we show that the switching lower-value function is a viscosity solution of the appropriate systems of quasi-variational inequalities(the appropriate generalization of the Hamilton-Jacobi equation in this context) and that the minimal such switching-storage function is equal to the continuous switching lower-value for the game. With the lower value function a optimal switching control is designed for minimizing the cost of running the systems.     

Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay

Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state.     

Weak topology and infinite matrix games

This paper approaches infinite matrix games through the weak topology on the players' sets of strategies. A new class of semi-infinite and infinite matrix games is defined, and it is proved that these games always have a value and optimal strategies for each player. Using these games it is proved that some other important classes of infinite matrix game also have values. Received April 1996/Revised version June 1997/Final version September 1997     

The gold-mine game

The paper considers the following two-person zero-sum game. The minimizing player chooses to hide his gold and a mine in two distinct boxes from an infinite number of boxes labelled 1, 2, 3,.... The maximizing player now chooses to open the boxes in some order, and if he finds the gold before the mine the payoff to him is 1; otherwise, the payoff is zero. The game is solved in the sense of Kindler.     

连续对策上的计策问题

限定一个连续对策不是平凡地无意义（例如对某个局中人绝对有利等），我们提出了连续对策上的计策的基本概念。最后得到结论，如果局中人1使用经典对策，那么他的赢得期望必不是赢得函数的最大值。如果局中人1使用计策成功（即使得局中人2中计），那么局中人1必取得赢得函数的最大值，局中人2也有对偶的结果。     

On multivariate antagonistic marked point processes

This paper investigates two bivariate “antagonistic” processes in the framework of noncooperative games that find applications in economics, engineering, and warfare. These processes represent two players with entirely opposite interests. Hostile actions, associating with one of the components (such as deliberate sellout of stocks of the opponent), take place at random times having a random effect to various vital areas. A consequence of these actions is material damage (such as financial losses) which is associated with the second component. So, in general, each player has two (dependent) ways of striking, while his opponent can sustain multiple damages (of two kinds and interdependent) until his threshold of endurance is reached. Then, the player is ruined. One of the goals is to predict the ruin time of each player as well as the collateral damage to the loser and winner at the ruin time. Furthermore, it is also of vital interest to relate these random elements to the continuous time parameter processes referred to as time sensitive analysis. We succeed in these goals by arriving at closed form joint functionals of the named elements and processes.     

Emergence and nonemergence of alternating offers in bilateral bargaining

We study in what circumstance players alternate offers in bilateral bargaining. To examine this question, we suppose that players choose whether to take the initiative in each period. The player who tries to take the initiative is able to make an offer only when the other player does not. The probability that a player tries to take the initiative is referred to as the frequency of initiative taking. We assume that this is conditioned on mutually observable states and is, once chosen, unchangeable. When players make their frequency of initiative taking dependent on the identity of the latest proposer, the players alternate their offers (possibly with some stochastic delay). In contrast, when players always use the same frequency of initiative taking, or when players only distinguish odd-numbered from even-numbered periods for the frequency of initiative taking, both players constantly try to take the initiative. Consequently, an impasse arises.