# 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 an applied mathematicsjournal that publishes high quality peer-review and expository papersin all research areas of expertise of its editors. The main focus ofJDG is in the interface of Dynamical Systems and Game Theory. Click here for more information

SUBMISSIONS TO REGULAR ISSUES
The submissions to JDG regular issues are done by email to the editors in chief or directly to one of the editors. The authors should ask for an email from the editors in chief confirming the reception of the paper.

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.

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

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2018, 5(2) : 61-94 doi: 10.3934/jdg.2018006 +[Abstract](315) +[HTML](286) +[PDF](721.98KB)
Abstract:

The present paper aims to study a robust-entropic optimal control problem arising in the management of financial institutions. More precisely, we consider an economic agent who manages the portfolio of a financial firm. The manager has the possibility to invest part of the firm's wealth in a classical Black-Scholes type financial market, and also, as the firm is exposed to a stochastic cash flow of liabilities, to proportionally transfer part of its liabilities to a third party as a means of reducing risk. However, model uncertainty aspects are introduced as the manager does not fully trust the model she faces, hence she decides to make her decision robust. By employing robust control and dynamic programming techniques, we provide closed form solutions for the cases of the (ⅰ) logarithmic; (ⅱ) exponential and (ⅲ) power utility functions. Moreover, we provide a detailed study of the limiting behavior, of the associated stochastic differential game at hand, which, in a special case, leads to break down of the solution of the resulting Hamilton-Jacobi-Bellman-Isaacs equation. Finally, we present a detailed numerical study that elucidates the effect of robustness on the optimal decisions of both players.

2018, 5(2) : 95-107 doi: 10.3934/jdg.2018007 +[Abstract](287) +[HTML](168) +[PDF](386.75KB)
Abstract:

In this paper we propose new games that satisfy nested constraints given by a level structure of cooperation. This structure is defined by a family of partitions on the set of players. It is ordered in such a way that each partition is a refinement of the next one. We propose a value for these games by adapting the Shapley value. The value is characterized axiomatically. For this purpose, we introduce a new property called class balance contributions by generalizing other properties in the literature. Finally, we introduce a multilinear extension of our games and use it to obtain an expression for calculating the adapted Shapley value.

2018, 5(2) : 109-141 doi: 10.3934/jdg.2018008 +[Abstract](287) +[HTML](195) +[PDF](599.26KB)
Abstract:

This paper deals with two-person nonzero-sum stochastic differential games (SDGs) with an additive structure, subject to constraints that are additive also. Our main objective is to give conditions for the existence of constrained Nash equilibria for the case of infinite-horizon discounted payoff. This is done by means of the Lagrange multipliers approach combined with dynamic programming arguments. Then, following the vanishing discount approach, the results in the discounted case are used to obtain constrained Nash equilibria in the case of long-run average payoff.

2018, 5(2) : 143-163 doi: 10.3934/jdg.2018009 +[Abstract](219) +[HTML](218) +[PDF](559.55KB)
Abstract:

This paper deals with the risk probability for finite horizon semi-Markov decision processes with loss rates. The criterion to be minimized is the risk probability that the total loss incurred during a finite horizon exceed a loss level. For such an optimality problem, we first establish the optimality equation, and prove that the optimal value function is a unique solution to the optimality equation. We then show the existence of an optimal policy, and develop a value iteration algorithm for computing the value function and optimal policies. We also derive the approximation of the value function and the rules of iteration. Finally, a numerical example is given to illustrate our results.

2018, 5(2) : 165-185 doi: 10.3934/jdg.2018010 +[Abstract](144) +[HTML](104) +[PDF](535.26KB)
Abstract:

In this article, we construct examples of discrete-time, dynamic, partial equilibrium, single product, competition market sequences, namely, \begin{document}$\{m_{t}\}_{t = 0}^{∞}$\end{document}, in which, potentially active firms, are countably infinite, the inverse demand function is linear, and the initial market \begin{document}$m_0$\end{document} is null. For Cournot markets, in which, the number of firms is defined exogenously, as a finite positive integer, namely \begin{document}$n: n>3$\end{document}, the long term behavior of the quantity supplied, into the market, by Cournot firms is not well explored and is unknown. In this article, we conjecture, that in all such cases, the Cournot equilibrium, provided that it exists, is unreachable. We construct Cournot market sequences, which might be viewed, as appropriate resource tools, through which, the "unreachability" of Cournot equilibrium points is being resolved. Our construction guidelines are, the stable manifolds of Cournot equilibrium points. Moreover, if the number of active firms, increases to infinity and the marginal costs of all active firms are identical, the aggregate market supply, increases to a competitive limit and each firm, at infinity, faces a market price equal to its marginal cost. Hence, the market sequence approaches a perfectly competitive equilibrium. In the case, where marginal costs are not identical, we show, that there exists a market sequence, \begin{document}$\{m_{t}\}_{t = 0}^{∞}$\end{document}, which approaches an infinite dimensional Cournot equilibrium point. In addition, we construct a sequence of Cournot market sequences, namely, \begin{document}$\{m_{it}\}_{t = 0}^{∞}, i ≥ 1$\end{document}, which, for each, \begin{document}$i$\end{document}, approaches an imperfectly competitive equilibrium. The sequence of the equilibrium points and the double sequence, \begin{document}$\{m_{it}\}$\end{document}, both approach, the equilibrium, at infinity, of the market sequence, \begin{document}$\{m_t\}$\end{document}.

2018, 5(2) : 187-187 doi: 10.3934/jdg.2018011 +[Abstract](124) +[HTML](84) +[PDF](133.0KB)
Abstract:
2017doi: 10.3934/jdg.2018012 +[Abstract](34) +[HTML](25) +[PDF](440.06KB)
2017doi: 10.3934/jdg.2018013 +[Abstract](56) +[HTML](87) +[PDF](431.57KB)
2014, 1(3) : 471-484 doi: 10.3934/jdg.2014.1.471 +[Abstract](362) +[PDF](385.9KB) Cited By(8)
2014, 1(3) : 363-375 doi: 10.3934/jdg.2014.1.363 +[Abstract](415) +[PDF](386.0KB) Cited By(6)
2014, 1(3) : 411-445 doi: 10.3934/jdg.2014.1.411 +[Abstract](313) +[PDF](550.1KB) Cited By(6)
2015, 2(1) : 65-87 doi: 10.3934/jdg.2015.2.65 +[Abstract](300) +[PDF](535.9KB) Cited By(4)
2015, 2(1) : 103-115 doi: 10.3934/jdg.2015.2.103 +[Abstract](295) +[PDF](364.9KB) Cited By(4)
2014, 1(2) : 181-254 doi: 10.3934/jdg.2014.1.181 +[Abstract](390) +[PDF](897.5KB) Cited By(4)
2014, 1(4) : 555-578 doi: 10.3934/jdg.2014.1.555 +[Abstract](461) +[PDF](527.0KB) Cited By(4)
2015, 2(3&4) : 257-287 doi: 10.3934/jdg.2015004 +[Abstract](371) +[PDF](538.5KB) Cited By(3)
2016, 3(4) : 371-398 doi: 10.3934/jdg.2016020 +[Abstract](443) +[PDF](831.3KB) Cited By(3)
2014, 1(1) : 121-151 doi: 10.3934/jdg.2014.1.121 +[Abstract](396) +[PDF](2417.1KB) Cited By(3)