Mean Residual Life Processes and Associated Submartingales

We use an argument of Madan and Yor to construct associated submartingales to a class of two-parameter processes that are ordered by increasing convex dominance. This class includes processes whose integrated survival functions are multivariate totally positive of order 2 (MTP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MTP}_2$$\end{document}). We prove that the integrated survival function of an integrable two-parameter process is MTP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MTP}_2$$\end{document} if and only if it is totally positive of order 2 (TP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TP}_2$$\end{document}) in each pair of arguments when the remaining argument is fixed. This result cannot be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. Since the MTP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MTP}_2$$\end{document} property is closed under several transformations, it allows us to exhibit many other processes having the same total positivity property.