Abstract
Modeling the interactions between a reinsurer and several insurers, or between a central management branch (CB) and several subsidiary business branches, or between a coalition and its members, are fascinating problems, which suggest many interesting questions. Beyond two dimensions, one cannot expect exact answers. Occasionally, reductions to one dimension or heuristic simplifications yield explicit approximations, which may be useful for getting qualitative insights. In this paper, we study two such problems: the ruin problem for a two-dimensional CB network under a new mathematical model, and the problem of valuation of two-dimensional CB networks by optimal dividends. A common thread between these two problems is that the one dimensional reduction exploits the concept of invariant cones. Perhaps the most important contribution of the paper is the questions it raises; for that reason, we have found it useful to complement the particular examples solved by providing one possible formalization of the concept of a multi-dimensional risk network, which seems to us an appropriate umbrella for the kind of questions raised here.
Highlights
Multi-dimensional risk networks is an exciting discipline which emerged recently—see for example Kriele and Wolf (2014);
The index set for the network is denoted by I = {0, 1, . . . , I }, with the index 0 being reserved for a central branch (CB), if such a branch exists
We introduce below a new reinsurance/central branch (CB) network model which combines the bail-out model of Avram and Minca (2015, 2017); Avram et al (2016) with an older model studied in
Summary
Multi-dimensional risk networks is an exciting discipline which emerged recently—see for example Kriele and Wolf (2014); We introduce below a new reinsurance/central branch (CB) network model which combines the bail-out model of Avram and Minca (2015, 2017); Avram et al (2016) with an older model studied in. This section illustrates the utility of our new combined model, since without adding sufficient proportional reinsurance, the Laplace transform could not be obtained analytically in the previous papers Avram and Minca (2015, 2017); Avram et al (2016). This section is included to illustrate the fact that heuristics may contribute, together with numerical methods, to unravel the mysteries of multi-dimensional risk networks, and the potential of applying the already available one and two-dimensional tools for managing multi-dimensional CB networks
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