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Mean Residual Life Processes and Associated Submartingales

Abstract

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 (\(\hbox {MTP}_2\)). We prove that the integrated survival function of an integrable two-parameter process is \(\hbox {MTP}_2\) if and only if it is totally positive of order 2 (\(\hbox {TP}_2\)) 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 \(\hbox {MTP}_2\) property is closed under several transformations, it allows us to exhibit many other processes having the same total positivity property.

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Acknowledgements

We are grateful to the anonymous referee for a careful reading and valuable comments that led to a substantial improvement of the paper.

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Correspondence to Antoine-Marie Bogso.

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Bogso, A. Mean Residual Life Processes and Associated Submartingales. J Theor Probab 33, 36–64 (2020). https://doi.org/10.1007/s10959-018-0865-6

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Keywords

  • Cox–Hobson algorithm
  • Incomplete Markov processes
  • MRL ordering
  • Two-parameter submartingales
  • Total positivity

Mathematics Subject Classification (2010)

  • 60E15
  • 60G44
  • 60J25
  • 32F17