ISSN:

2164-6066

eISSN:

2164-6074

## Journal of Dynamics & Games

April 2018 , Volume 5 , Issue 2

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**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.

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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.

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**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.

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**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.

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**Abstract:**

In this article, we construct examples of discrete-time, dynamic, partial equilibrium, single product, competition market sequences, namely,

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