Volume 18, 2014
|Page(s)||514 - 540|
|Published online||10 October 2014|
An application of multivariate total positivity to peacocks
1 Universitéde Lorraine, Institut Elie
Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, 54506, France
2 CNRS, Institut Elie Cartan de Lorraine, UMR 7502, 54506, Vandoeuvre-lès-Nancy, France
Revised: 14 May 2013
We use multivariate total positivity theory to exhibit new families of peacocks. As the authors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associated martingales vol. 3. Bocconi-Springer (2011)], our guiding example is the result of Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao, Finance Res. Lett. 5 (2008) 162–171]. We shall introduce the notion of strong conditional monotonicity. This concept is strictly more restrictive than 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 are many random vectors which are strongly conditionally monotone (SCM). Indeed, we shall prove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As a consequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This family includes processes with independent and log-concave increments, and one-dimensional diffusions which have absolutely continuous transition kernels.
Mathematics Subject Classification: 60J25 / 32F17 / 60G44 / 60E15
Key words: Convex order / peacocks / total positivity of order 2 (TP2) / multivariate total positivity of order 2 (MTP2) / markov property / strong conditional monotonicity
© EDP Sciences, SMAI 2014
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