Hessian and increasing-Hessian orderings of multivariate skew-elliptical random vectors with applications in actuarial science

In this work, we establish some stochastic comparison results for multivariate skew-elliptical random vectors. These multivariate stochastic comparisons involve Hessian and increasing-Hessian orderings and many of their special cases. We provide necessary and/or sufficient conditions for the orderings by comparing the underlying model parameters. In addition, we investigate the (positive) linear forms of usual stochastic, convex and increasing convex, positive convex and increasing-positive-convex orderings. Using these theoretical results, we explore two applications. The first involves determining the upper bound of multivariate skew-elliptical risk variables under specific parameter constraints. The other one focuses on assessing the portfolio aggregation risks. Finally, two examples based on numerical simulations and real data from an Australian insurance company illustrate the established results and practical explanations.