The restricted convex risk measures in actuarial solvency

In this article, we propose a class of convex risk measures defined on appropriate wedges of a space of financial positions which denote the cumulative surplus variables created by undertaking risks by either an insurance or a reinsurance company. The form of the wedge which is the domain of such a risk measure expresses the form of the company, and it is a subspace in the case of reinsurance companies and a cone in the case of the insurance companies. The value of such a risk measure on an insurance position denotes the capital that the corresponding company has to receive or to keep in advance so that it will not be exposed to risk due to this position. We prove some dual representation and continuity results being similar to the unrestricted case. Finally, we contribute to a decision theory related to the choice of a numeraire asset when the space in which the positions lie in is reflexive. Copyright Springer-Verlag 2014