Main Contributions and Outline. In the first part of this thesis, we solve two general optimal reinsurance problems in Chapter 2. The first one is the minimization of a general functional of the ex- pectation, Value-at-Risk, and Tail Value-at-Risk of the total retained loss of the insurer while the premium principle determined by the reinsurer is a convex order preserving functional, together with the budget constraint. To solve this first gen- eral problem, instead of using the classical Xxxxxx-Xxxxxxxx once-crossing criterion as in the literature, we apply a generalization of it, namely, the Xxxxxx-Xxxxxxxx- Xxxxxx-Xxxxxx (multiple) crossing conditions. To the best of our knowledge, this is the first application of the multiple crossing conditions to solve optimal rein- surance problems. Our results in the first problem are generalization of those in Chi (2012b). The second problem is the minimization of a general law-invariant continuous from above coherent risk measure of the total retained loss of the in- surer while the premium principle determined by the reinsurer is a law-invariant comonotonic additive convex risk measure, together with the budget constraint. To solve this second general problem, we first represent the objective function and the premium principle in terms of distortion functions; we then apply the mini- max theorem for infinite dimensional space to interchange the infimum on the space of indemnities and the supremum on the space of probability measures; we finally employ the recent results by Xx (2017) to solve the minimization problem. Our results in the second problem can be regarded as a partial generalization of Xxxxxx et al. (2014), in which they minimized a general law-invariant convex risk measure but with the expected value premium principle and without the budget constraint. In Chapter 3, the second part of this thesis, we study the forward entropic risk measures, first introduced by Xxxxxxxxxxxxx and Xˇxxxxxx´c (2010). The for- xxxx entropic risk measures, similar to the classical entropic risk measures, are defined in a decision theoretic framework using the indifference pricing approach. Moreover, the forward entropic risk measures are maturity independent risk mea- sures, which were first introduced in Zariphopoulou and Zˇitkovi´c (2010) to rem- edy the deficiency in the classical dynamic risk measures. These motivated us to further investigate the behavior of the forward entropic risk measures. The im- portant ingredient in the forward entropic ri...
