Lebesgue property of convex risk measures for bounded Càdlàg processes

Abstract

In this paper, we study the Lebesgue property for convex risk measures on the class of bounded càdlàg processes. For that, we characterize the compact subsets of a family of bounded variation processes, which is, of course, the topological dual of the bounded càdlàg processes, in an appropriate topology. We show that the Lebesgue property can be characterized in several equivalent ways.

On the Capital Allocation Problem for a New Coherent Risk Measure in Collective Risk Theory

“…A comprehensive understanding of the problem of capital allocation for coherent risk measures in this setting requires advanced notions and techniques from functional analysis as well as convex analysis. In fact, a formal treatment of the problem of capital allocation for coherent risk measures requires studying the weak sub-gradient set associated to the risk measure (see [30]). As it turns out, in order to get a good understanding of the sub-gradient set and its properties, we need a robust representation of the underlying risk measure which, in turn, requires studying of the dual space of R p L .…”

“…To the best of our knowledge, this problem has not been thoroughly studied for risk measures defined on the space of stochastic processes. We can only cite, [30], where the author discusses the problem of capital allocation for risk measures defined on the space of cádlág processes. Or, [6] where the authors study the capital allocation problem for a new risk measure that, as it turned out, it does not satisfy an axiomatic definition of coherent risk measures defined on stochastic processes proposed in [8].…”

“…Such constructions were proposed and studied in [30]. The features of such measures inherently depend on the choice of base risk measure ρ 0 .…”

“…Since s * is the solution of setting Equation (30) equal to zero, we can simplify (31) as follows, for i = 1, . .…”

“…In others, such an explicit form is not available but a solution can still be obtained by standard numerical methods. The difficulty lies in solving Equation (30) when is set to be equal to zero. We now discuss some particular models that have appeared in the literature on insurance Lévy risk processes and give explicit and semi-explicit expressions for their capital allocation.…”