Hitting probabilities for compound Poisson processes in a bipartite network

Anita Behme, Claudia Klüppelberg, Gesine Reinert

Published 2018-05-31Version 1

This paper studies hitting probabilies of a constant barrier for single and multiple components of a multivariate compound Poisson process. The components of the process are allowed to be dependent, with the dependency structure of the components induced by a random bipartite network. In analogy with the non-network scenario, a network Pollaczek-Khintchine parameter $P$ is introduced. This random parameter, which depends on the bipartite network, is crucial for the hitting probabilities. Under certain conditions on the network and for light-tailed jump distributions we obtain Lundberg bounds and, for exponential jump distributions, exact results for the hitting probabilities. For large sparse networks, the parameter $P$ is approximated by a function of independent Poisson variables. As applications, risk balancing networks in ruin theory and load balancing networks in queuing theory are discussed.