An application of multivariate total positivity to peacocks

Bogso, Antoine Marie

ESAIM: Probability and Statistics, Tome 18 (2014), pp. 514-540.

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.

Classification : 60J25, 32F17, 60G44, 60E15
Mots clés : convex order, peacocks, total positivity of order 2 (TP2), multivariate total positivity of order 2 (MTP2), markov property, strong conditional monotonicity

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