An application of multivariate total positivity to peacocks

TL;DR

This paper applies multivariate total positivity theory to identify new families of peacocks, introducing the concept of strong conditional monotonicity and demonstrating its relation to MTP2 random vectors and certain stochastic processes.

Contribution

It introduces the notion of strong conditional monotonicity and shows that MTP2 random vectors are SCM, expanding the class of processes known to exhibit peacock properties.

Findings

MTP2 random vectors are strongly conditionally monotone.

Processes with MTP2 marginals include those with independent, log-concave increments.

Diffusions with absolutely continuous transition kernels are SCM.

Abstract

We use multivariate total positivity theory to exhibit new families of peacocks. As the authors of \cite{HPRY}, our guiding example is the result of Carr-Ewald-Xiao \cite{CEX}. We shall introduce the notion of strong conditional monotonicity. This concept is strictly more restrictive than the conditional monotonicity as defined in \cite{HPRY} (see also \cite{Be}, \cite{BPR1} and \cite{ShS1}). There are many random vectors which are strongly conditionally monotone (SCM). Indeed, we shall prove that multivariate totally positive of order 2 (MTP$_2$) random vectors are SCM. As a consequence, stochastic processes with MTP$_2$ finite-dimensional marginals are SCM. This family includes processes with independent and log-concave increments, and one-dimensional diffusions which have absolutely continuous transition kernels.