An application of multivariate total positivity to peacocks

Abstract

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.