On Haezendonck risk measures

We study the Haezendonck risk measure (introduced by [Haezendonck, J., Goovaerts, M., 1982. A new premium calculation principle based on Orlicz norms. Insurance: Mathematics and Economics 1, 41-53] and by [Goovaerts, M.J., Kaas, R., Dhaene, J., Tang, Q., 2003. A unified approach to generate risk measures. ASTIN Bulletin 33 (2), 173-191; Goovaerts, M.J., Kaas, R., Dhaene, J., Tang, Q., 2004. Some new classes of consistent risk measures. Insurance: Mathematics and Economics 34 (3), 505-516]) and prove its subadditivity. Since the Haezendonck risk measure is defined as an infimum of Orlicz premia, we investigate when the infimum is actually attained. We determine the corresponding generalized scenarios and show how its construction can be seen as a special case of the operation of inf-convolution of convex functionals.