Mean residual life processes and associated submartingales

TL;DR

This paper explores the construction of submartingales associated with two-parameter processes, establishing conditions for MTP$_2$ properties and their implications for mean residual life processes.

Contribution

It introduces a novel method using Madan-Yor's argument to connect submartingales with two-parameter processes and characterizes MTP$_2$ properties in this context.

Findings

Integrated survival functions are MTP$_2$ iff TP$_2$ in each fixed argument.

Certain MRL processes possess the MTP$_2$ property, enabling the identification of related processes.

The results extend known properties to processes with non-interval support.

Abstract

We use Madan-Yor's argument to construct associated submartingales to a class of two-parameter processes that are ordered by the increasing convex dominance. This class includes processes which have MTP$_2$ integrated survival functions. We prove that the integrated survival function of an integrable two-parameter process is MTP$_2$ if and only if it is TP$_2$ in each pair of arguments when the remaining argument is fixed. This result can not be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. The MTP$_2$ property of certain MRL processes is useful to exhibit numerous other processes having the same property.