Stochastic monotonicity and the Markov product for copulas

TL;DR

This paper characterizes stochastically monotone copulas using Markov operators and the Markov product, identifying all idempotent cases as ordinal sums of independence copulas.

Contribution

It introduces two novel characterizations of stochastically monotone copulas via Markov operators and the Markov product, linking copula theory with Markov process properties.

Findings

Established a one-to-one correspondence between stochastically monotone copulas and monotonicity-preserving Markov operators.

Characterized stochastically monotone copulas by their monotonicity with respect to the Markov product.

Identified all idempotent stochastically monotone copulas as ordinal sums of the independence copula.

Abstract

Given two random variables $X$ and $Y$, stochastic monotonicity describes a monotone influence of $X$ on $Y$. We prove two different characterizations of stochastically monotone $2$-copulas using the isomorphism between $2$-copulas and Markov operators. The first approach establishes a one-to-one correspondence between stochastically monotone copulas and monotonicity-preserving Markov operators. The second approach characterizes stochastically monotone copulas by their monotonicity property with respect to the Markov product. Applying the latter result, we identify all idempotent stochastically monotone copulas as ordinal sums of the independence copula $\Pi$.