A Dynamic Contagion Risk Model with Recovery Features

We introduce threshold growth in the classical threshold contagion model, or equivalently a network of Cramér-Lundberg processes in which nodes have downward jumps when there is a failure of a neighboring node. Choosing the configuration model as underlying graph, we prove fluid limits for the baseline model, as well as extensions to the directed case, state-dependent interarrival times and the case of growth driven by upward jumps. We obtain explicit ruin probabilities for the nodes according to their characteristics: initial threshold and in- (and out-) degree. We then allow nodes to choose their connectivity by trading off link benefits and contagion risk. We define a rational equilibrium concept in which nodes choose their connectivity according to an expected failure probability of any given link and then impose condition that the expected failure probability coincides with the actual failure probability under the optimal connectivity. We show existence of an asymptotic equilibrium and convergence of the sequence of equilibria on the finite networks. In particular, our results show that systems with higher overall growth may have higher failure probability in equilibrium.

Social Learning Equilibria

We consider a large class of social learning models in which a group of agents face uncertainty regarding a state of the world, share the same utility function, observe private signals, and interact in a general dynamic setting. We introduce social learning equilibria, a static equilibrium concept that abstracts away from the details of the given extensive form, but nevertheless captures the corresponding asymptotic equilibrium behavior. We establish general conditions for agreement, herding, and information aggregation in equilibrium, highlighting a connection between agreement and information aggregation.

Class of controllable systems of differential equations for infinite time

In the article necessary conditions for a controllability of systems of nonlinear differential equations in an infinite time are obtained without assuming the existence of an asymptotic equilibrium for the system of linear approximation. Thus, a new class of controlled systems of differential equations is presented. The problem of controllability for an infinite time (i.e. the transfer of an arbitrary point into an arbitrary small domain of another point) comes down to choosing an operator depending on the selected control, which in turn depends on the point being transferred. Then one is to prove the existence of a fixed point for this operator. It is known that the theorems on controllability require existence of an asymptotic equilibrium for system of the first approximation. It is shown in the paper that in general case the condition of asymptotic equilibrium’s existence is not necessary for controllability of systems in an infinite time. An example on the theorem on controllability for an infinite time is given. The theorem generalizing Vazhevsky inequality is proved by implementation of Cauchy-Bunyakovsky inequality. A remark is made about the theorem’s validity for the case when the matrix and vector from the right-hand side of nonlinear differential equation are complex and x is vector with complex components. Basing on the left-hand side of the inequality in the theorem generalizing Vazhevsky inequality, the necessary conditions for controllability in an infinite time are obtained. These conditions are verified on the same example of a scalar equation that was mentioned before.

Numerical Simulation of Entropy Growth for a Nonlinear Evolutionary Model of Random Markets

In this communication, the generalized continuous economic model for random markets is revisited. In this model for random markets, agents trade by pairs and exchange their money in a random and conservative way. They display the exponential wealth distribution as asymptotic equilibrium, independently of the effectiveness of the transactions and of the limitation of the total wealth. In the current work, entropy of mentioned model is defined and then some theorems on entropy growth of this evolutionary problem are given. Furthermore, the entropy increasing by simulation on some numerical examples is verified.

A Note on Asymptotic Equilibrium for Fuzzy Differential Equations

The asymptotic equilibrium results for fuzzy differential systemsGCP x′=f1t,x,y,y′=f2(t,x,y)are investigated, wheref1t,x,ysatisfies the compactness-type andf2t,x,ysatisfies the dissipative-type conditions. It is worth mentioning that the uniformly continuous conditions offt,x,yare removed in Song et al. (2005). That is to say, the results of Song et al. (2005) are extended. In addition, the global existence and asymptotic equilibrium results of fuzzy differential systemsCP x′t=ft,x,x0=x0are obtained.