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An application of multivariate total positivity topeacocks

Published online by Cambridge University Press:  10 October 2014

Antoine Marie Bogso
Antoine Marie Bogso*
Affiliation:
Universitéde Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, 54506, FranceCNRS, Institut Elie Cartan de Lorraine, UMR 7502, 54506, Vandoeuvre-lès-Nancy, France. antoine.bogso@univ-lorraine.fr; ambogso@gmail.com

Abstract

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We use multivariate total positivity theory to exhibit new families of peacocks. As theauthors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associatedmartingales vol. 3. Bocconi-Springer (2011)], our guiding example is the resultof Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao,Finance Res. Lett. 5 (2008) 162–171]. We shall introducethe notion of strong conditional monotonicity. This concept is strictly more restrictivethan the conditional monotonicity as defined in [F. Hirsch, C. Profeta, B. Roynette and M.Yor, Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011)] (see also [R.H.Berk, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42(1978) 303–307], [A.M. Bogso, C. Profeta and B. Roynette, Lect. Notes Math.Springer, Berlin (2012) 281–315.] and [M. Shaked and J.G. Shanthikumar,Probab. Math. Statistics. Academic Press, Boston (1994)].). There aremany random vectors which are strongly conditionally monotone (SCM). Indeed, we shallprove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As aconsequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This familyincludes processes with independent and log-concave increments, and one-dimensionaldiffusions which have absolutely continuous transition kernels.

Keywords

Convex orderpeacockstotal positivity of order 2 (TP2)multivariate total positivity of order 2 (MTP2)markov propertystrong conditional monotonicity

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014, pp. 514 - 540
Copyright
© EDP Sciences, SMAI 2014

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