Extremal stochastic integrals: a parallel . . .

We construct extremal stochastic integrals Re E f(u)M®(du) of a deterministic function f(u) ¸ 0 with respect to a random ®¡Fr¶echet ( ®> 0) sup{measure. The measure M® is sup{additive rather than additive and is defined over a general measure space (E; E; ), where is a deterministic control measure. The extremal integral is constructed in a way similar to the usual ®¡stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary f(u) ¸ 0; R E f(u)®(du) <1, and the metric ®(f; g):= R E jf(u)®¡g(u)®j(du) metrizes the convergence in probability of the resulting integrals. This approach complements the well-known de Haan's spectral representation of max{ stable processes with ®¡Fr¶echet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of ®¡stable processes, whereas the extremal integrals correspond to the usual ®¡stable stochastic integrals. We prove that essentially any strictly ®¡stable process belongs to the domain of max{stable attraction of an ®¡Fr¶echet, max-stable process. Moreover, we express the corresponding ®¡Fr¶echet processes in terms of extremal stochastic integrals, involving the kernel function of the ®¡stable process. The close correspondence between the max-stable and ®¡stable frameworks yields new examples of max-stable processes with non-trivial dependence structures