An application of multivariate total positivity to peacocks. (10th October 2014)

• Record Type: Journal Article
• Authors: Bogso, Antoine Marie
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 514
• Page End: 540
• Publication Date: 2014-10-10
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013049
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>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, <italic>Peacocks and associated martingales </italic>vol. 3. Bocconi-Springer (2011)], our guiding example is the result of Carrâˆ'Ewaldâˆ'Xiao [P. Carr, C.-O. Ewald and Y. Xiao, <italic>Finance Res. Lett. </italic><bold>5 </bold>(2008) 162â€"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, <italic>Z. Wahrscheinlichkeitstheorie und Verw. Gebiete </italic><bold>42 </bold>(1978) 303â€"307], [A.M. Bogso, C. Profeta and B. Roynette, <italic>Lect. Notes Math. </italic>Springer, Berlin (2012) 281â€"315.] and [M. Shaked and J.G. Shanthikumar, <italic>Probab. Math. Statistics</italic>. 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 (MTP<sub>2</sub>) random vectors are SCM. As a consequence, stochastic processes with MTP<sub>2</sub> 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.</p> </abstract>