Ruin Probabilities for Risk Processes in Stochastic Networks

TL;DR

This paper analyzes ruin probabilities in large stochastic networks with agents receiving losses from neighbors, providing explicit formulas for infinite network limits and bounds for complex open networks.

Contribution

It introduces a classification approach to reduce dimensionality and derives explicit ruin probabilities for agents in large-scale stochastic networks.

Findings

Explicit ruin probabilities for agents as network size tends to infinity

Dependence of ruin probabilities on network structure, agent types, and loss processes

Bounds on ruin probabilities for open networks with external losses

Abstract

We study multidimensional Cram\'er-Lundberg risk processes where agents, located on a large sparse network, receive losses form their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities.