Interval-based stochastic dominance: theoretical framework and application to portfolio choices

We introduce a new stochastic dominance relationship, the interval-based stochastic dominance (ISD). By choosing different reference points, we show that ISD may span a continuum of preferences between kth and (k+ 1 ) th order stochastic dominance (SD). We distinguish accordingly between interval-based (or shortly just interval) SD of order 1 and of order 2: the former spanning from first- to second-order stochastic dominance, the latter from second- to third-order stochastic dominance. By examining the relationships between interval-based SD and SD, as well as between ISD and risk measures or utility functions, we frame the concept within decision theory and clarify its implications when applied to an optimal financial allocation problem. The formulation of ISD-constrained problems in the presence of discrete random variables is discussed in detail and applied to a portfolio selection problem.