MRL order, log-concavity and an application to peacocks

TL;DR

This paper establishes a new log-concavity condition equivalent to the MRL order for real-valued processes, enabling the construction of new MRL-increasing processes and linking peacocks to explicit martingale embeddings.

Contribution

It introduces an equivalent log-concavity criterion for MRL ordering and constructs new MRL processes with applications to peacocks and martingale embeddings.

Findings

New families of MRL-increasing processes identified

Log-concavity condition equivalent to MRL order established

Explicit martingale constructions for peacocks provided

Abstract

We provide an equivalent log-concavity condition to the mean residual life (MRL) ordering for real-valued processes. This result, combined with classical properties of total positivity of order 2, allows to exhibit new families of integrable processes which increase in the MRL order (MRL processes). Note that MRL processes with constant mean are peacocks to which the Az\'ema-Yor (Skorokhod embedding) algorithm yields an explicit associated martingale.