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.
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 sets of risk measures. In this study, we extend the inf-convolution of risk measures in its convex-combination form to a countable (not necessarily finite ) set of alternatives. The intuitive principle of this approach 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.
D
(or at most countable in rare cases)
R
an arbitrary
a countable
D
or even countable
D
is to regard a probability measure as