Stochastic order

A stochastic order is an order on probability measures induced by a given order on the underlying space. Since there are many different ways to extend the order from the base space to probability distributions on the space, there are many different stochastic orders. Some authors speak of a “usual stochastic order,” under which one random variable is less than or equal to another whenever the corresponding survival functions are.

Literature

General

  • Shaked & Shanthikumar, 2007: Stochastic orders (doi)
    • The standard reference, an effort to “put in one place essentially all that is known about these orders”
    • Updated version of: Shaked & Shanthikumar, 1994: Stochastic orders and their applications
  • Müller & Stoyan, 2002: Comparison methods for stochastic models and risks

Category theory

References from the categorical probability literature:

  • Perrone, 2018, PhD thesis: Categorical probability and stochastic dominance in metric spaces (pdf)
  • Fritz & Perrone, 2020: Stochastic order on metric spaces and the ordered Kantorovich monad (doi, arxiv)