Inf-convolution and optimal risk sharing with countable sets of risk measures

Abstract

The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite (or at most countable in rare cases) sets of risk measures. In this study, we extend the inf-convolution of risk measures in its convex-combination form to an arbitrary (not necessarily finite or even countable) set of alternatives. The intuitive principle of this approach is to regard a probability measure as a generalization of convex weights in the finite case. Subsequently, we extensively generalize known properties and results to this framework. Specifically, we investigate the preservation of properties, dual representations, optimal allocations, and self-convolution.

Minkowski gauges and deviation measures

“…This concept is closely related to inf-convolution and optimal risk sharing. Inf-convolution is a well known operation for functionals in convex analysis -for details of the use of inf-convolution in risk share we refer the reader to Barrieu and El Karoui (2005), Jouini et al (2008) and Righi (2020a).…”