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.