Some new notions of dependence with applications in optimal allocation problems

On capital allocation for stochastic arrangement increasing actuarial risks

“…As a direct consequence of Theorem 4.1, we reach Proposition 5.1, which is exactly Theorem 6.5 of [2], (T , · · · , Tn) is SAD if and only if (e −T , · · · , e −Tn ) is SAI.…”

“…Recently, [2,3] introduced the following characterization for the monotonicity of mutually dependent random variables, extending the joint likelihood ratio order to multiple random variables.…”

“…Because the assumption of independence among risks is hardly realistic and the comonotonicity among risks corresponds to the extreme case, it is of interest to introduce more general dependence structure to concerned risks and thus make the model more exible in practice. Recently, for multiple risks [2] took the rst to introduce the stochastic arrangement increasing (SAI) (see De nition 2.3) property, which, as a generalization of the arrangement increasing function, is rather convenient in risk management due to integrating monotonicity with dependence. Subsequently, [3] and [23] addressed more detailed properties of SAI along with some applications to insurance and economics.…”

Properties of Stochastic Arrangement Increasing and Their Applications in Allocation Problems

“…To overcome this restriction, Shanthikumar and Yao (1991) introduced the notion of joint likelihood ratio orders, which incorporates dependence structure into stochastic comparison. Following this pioneering work, Cai and Wei (2014) and Cai and Wei (2015) generalized the notion of joint likelihood ratio order and developed the concepts of stochastic arrangement increasing (SAI) and weakly stochastically arrangement increasing through right and left tails (RWSAI; LWSAI). They further explored the applications of these concepts in the study of the allocations of deductibles and policy limits as well as portfolio selection problems.…”

Compositional methods applied to capital allocation problems

“…Previous studies have focused on particular probability distributions of losses (Cossette et al, 2012(Cossette et al, , 2013, on alternative dependence structures between risks (Cai and Wei, 2014), on asymptotic allocations based on commonly used risk measures (Asimit et al, 2011) or on optimization function alterations in order to overcome limitations of the loss function minimization allocation criterion (Xu and Hu, 2012;Xu and Mao, 2013). An exhaustive list is not provided here but it is the object of ongoing research.…”