Saddle-Type Functionals for Continuous Processes with Applications to Tests for Stochastic Spanning

We derive the continuity properties of the CDF of a random variable defined as a saddle-type point of a real valued continuous stochastic process on a compact metric space. This result facilitates the derivation of first order asymptotic properties of tests for stochastic spanning w.r.t. some stochastic dominance relation based on subsampling. As an illustration we define the concept of Markowitz stochastic spanning, derive an analytic representation upon the empirical analog of which we construct a relevant statistical test. The aforementioned result enables derivation of asymptotic exactness for the relevant procedure based on subsampling, when the metric space has the form of a simplicial complex, the spanning set is a compact subset and the significance level is chosen according to the number of extreme points of the complex inside the spanning set. Consistency is also derived. Such tests are of interest in financial economics since they can provide reductions of portfolio sets