# American Institute of Mathematical Sciences

ISSN:
2164-6066

eISSN:
2164-6074

## Journal Home

All Issues

### Volume 1, 2014

The Journal of Dynamics and Games (JDG) is a pure and applied mathematical journal that publishes high quality peer-review and expository papers in all research areas of expertise of its editors. The main focus of JDG is in the interface of Dynamical Systems and Game Theory. Click here for more information

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JDG Flyer

• AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
• Publishes 4 issues a year in January, April, July and October.
• Publishes online only.
• Indexed in Emerging Sources Citation Index, MathSciNet and Zentralblatt MATH.
• Archived in Portico and CLOCKSS.

“At this time of the passing of Professor Martin Shubik—one of Yale SOM’s most accomplished faculty members and a superbly accomplished contributor to the fields of mathematics and economics—we offer our deepest sympathies to his family and large numbers of friends and colleagues,” said Dean Edward A. Snyder.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

JDG Flyer

• AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
• Publishes 4 issues a year in January, April, July and October.
• Publishes online only.
• Indexed in Emerging Sources Citation Index, MathSciNet and Zentralblatt MATH.
• Archived in Portico and CLOCKSS.

“At this time of the passing of Professor Martin Shubik—one of Yale SOM’s most accomplished faculty members and a superbly accomplished contributor to the fields of mathematics and economics—we offer our deepest sympathies to his family and large numbers of friends and colleagues,” said Dean Edward A. Snyder.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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2019, 6(1) : 1-17 doi: 10.3934/jdg.2019001 +[Abstract](179) +[HTML](115) +[PDF](433.41KB)
Abstract:

We consider the minimax impulse control problem in finite horizon, when the cost functions are positive and not bounded from below with a strictly positive constant. We show existence of value function of the problem. Moreover, the value function is characterized as the unique viscosity solution of Hamilton-Jacobi-Bellman-Isaacs equation. This problem is in relation with an application in mathematical finance.

2019, 6(1) : 19-25 doi: 10.3934/jdg.2019002 +[Abstract](158) +[HTML](82) +[PDF](320.63KB)
Abstract:

Avoiding usual completeness hipothesis and working on the frame of locally complete spaces some Pareto optimization results are obtained. The Mackey Bishop-Phelps cones are defined and a characterization for the existence of Pareto efficiency respect to these cones is obtained.

2019, 6(1) : 27-51 doi: 10.3934/jdg.2019003 +[Abstract](192) +[HTML](159) +[PDF](529.79KB)
Abstract:

We study some ergodicity property of zero-sum stochastic games with a finite state space and possibly unbounded payoffs. We formulate this property in operator-theoretical terms, involving the solvability of an optimality equation for the Shapley operators (i.e., the dynamic programming operators) of a family of perturbed games. The solvability of this equation entails the existence of the uniform value, and its solutions yield uniform optimal stationary strategies. We first provide an analytical characterization of this ergodicity property, and address the generic uniqueness, up to an additive constant, of the solutions of the optimality equation. Our analysis relies on the theory of accretive mappings, which we apply to maps of the form $Id - T$ where $T$ is nonexpansive. Then, we use the results of a companion work to characterize the ergodicity of stochastic games by a geometrical condition imposed on the transition probabilities. This condition generalizes classical notion of ergodicity for finite Markov chains and Markov decision processes.

2019, 6(1) : 53-64 doi: 10.3934/jdg.2019004 +[Abstract](258) +[HTML](96) +[PDF](382.06KB)
Abstract:

In this work, through stochastic optimal control in continuous time the optimal decision making in consumption and investment is modeled by a rational economic agent, representative of an economy, who is a consumer and an investor adverse to risk; this in a finite time horizon of stochastic length. The assumptions of the model are: a consumption function of HARA type, a representative company that has a stochastic production process, the agent invests in a stock and an American-style Asian put option with floating strike equal to the geometric average subscribed on the stock, both modeled by controlled Markovian processes; as well as the investment of a principal in a bank account. The model is solved with dynamic programming in continuous time, particularly the Hamilton-Jacobi-Bellman PDE is obtained, and a function in separable variables is proposed as a solution to set the optimal trajectories of consumption and investment. In the solution analysis is determined: in equilibrium, the process of short interest rate that is driven by a square root process with reversion to the mean; and through a system of differential equations of risk premiums, a PDE is deduced equivalent to the Black-Scholes-Merton but to value an American-style Asian put option.

2019, 6(1) : 65-85 doi: 10.3934/jdg.2019005 +[Abstract](180) +[HTML](139) +[PDF](504.26KB)
Abstract:

Studying continuous time counterpart of some discrete time dynamics is now a standard and fruitful technique, as some properties hold in both setups. In game theory, this is usually done by considering differential games on Euclidean spaces. This allows to infer properties on the convergence of values of a repeated game, to deal with the various concepts of approachability, etc. In this paper, we introduce a specific but quite abstract differential game defined on the Wasserstein space of probability distributions and we prove the existence of its value. Going back to the discrete time dynamics, we derive results on weak approachability with partial monitoring: we prove that any set satisfying a suitable compatibility condition is either weakly approachable or weakly excludable. We also obtain that the value for differential games with nonanticipative strategies is the same that those defined with a new concept of strategies very suitable to make links with repeated games.

2014, 1(3) : 471-484 doi: 10.3934/jdg.2014.1.471 +[Abstract](1166) +[PDF](385.9KB) Cited By(8)
2014, 1(3) : 363-375 doi: 10.3934/jdg.2014.1.363 +[Abstract](1256) +[PDF](386.0KB) Cited By(6)
2014, 1(3) : 411-445 doi: 10.3934/jdg.2014.1.411 +[Abstract](1112) +[PDF](550.1KB) Cited By(6)
2015, 2(1) : 65-87 doi: 10.3934/jdg.2015.2.65 +[Abstract](1149) +[PDF](535.9KB) Cited By(4)
2015, 2(1) : 103-115 doi: 10.3934/jdg.2015.2.103 +[Abstract](1117) +[PDF](364.9KB) Cited By(4)
2014, 1(2) : 181-254 doi: 10.3934/jdg.2014.1.181 +[Abstract](1318) +[PDF](897.5KB) Cited By(4)
2014, 1(4) : 555-578 doi: 10.3934/jdg.2014.1.555 +[Abstract](1352) +[PDF](527.0KB) Cited By(4)
2015, 2(3&4) : 257-287 doi: 10.3934/jdg.2015004 +[Abstract](1266) +[PDF](538.5KB) Cited By(3)
2016, 3(4) : 371-398 doi: 10.3934/jdg.2016020 +[Abstract](1372) +[PDF](831.3KB) Cited By(3)
2014, 1(1) : 121-151 doi: 10.3934/jdg.2014.1.121 +[Abstract](1367) +[PDF](2417.1KB) Cited By(3)