On the optimal reinsurance problem (Q2866792)

The authors consider problems of the following general form NEWLINE\[NEWLINE \mathop{\arg\min}\limits_{R}\varrho(X-R+\pi(R)), NEWLINE\]NEWLINE where \(\varrho:L^p(\Omega,\mathcal{F},\mathbb{P})\rightarrow[0,\infty]\) is a risk functional, i.e. a convex, proper, normed, monotone (with respect to the almost sure order), lower semicontinuous and subdifferentiable mapping. The initial loss \(X\) is a random variable from \(L^p_+(\Omega,\mathcal{F},\mathbb{P})\), and \(R\in L^p\) is a suitable insurance coverage such that the residual loss minimizes their risk. The insurance coverage is provided by one insurer, who charges the insurance taker a premium according to a pricing rule \(\pi\). The authors consider the insurance coverage problem in two cases. In the first case, there are one insurance taker and one insurer, while in the second case, there are several insurance takers and one insurer.