Title: Spatial risk measures and rate of spatial diversification,
Abstract: An accurate assessment of the risk of extreme environmental events is of
great importance for populations, authorities and the banking/insurance
industry. Koch (2017) introduced a notion of spatial risk measure and a
corresponding set of axioms which are well suited to analyze the risk due to
events having a spatial extent, precisely such as environmental phenomena. The
axiom of asymptotic spatial homogeneity is of particular interest since it
allows one to quantify the rate of spatial diversification when the region
under consideration becomes large. In this paper, we first investigate the
general concepts of spatial risk measures and corresponding axioms further. We
also explain the usefulness of this theory for the actuarial practice. Second,
in the case of a general cost field, we especially give sufficient conditions
such that spatial risk measures associated with expectation, variance,
Value-at-Risk as well as expected shortfall and induced by this cost field
satisfy the axioms of asymptotic spatial homogeneity of order 0, -2, -1 and -1,
respectively. Last but not least, in the case where the cost field is a
function of a max-stable random field, we mainly provide conditions on both the
function and the max-stable field ensuring the latter properties. Max-stable
random fields are relevant when assessing the risk of extreme events since they
appear as a natural extension of multivariate extreme-value theory to the level
of random fields. Overall, this paper improves our understanding of spatial
risk measures as well as of their properties with respect to the space variable
and generalizes many results obtained in Koch (2017).