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.
Keywords: convex order, peacocks, total positivity of order 2 (TP2), multivariate total positivity of order 2 (MTP2), markov property, strong conditional monotonicity
@article{PS_2014__18__514_0,
author = {Bogso, Antoine Marie},
title = {An application of multivariate total positivity to peacocks},
journal = {ESAIM: Probability and Statistics},
pages = {514--540},
publisher = {EDP Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013049},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2013049/}
}
TY - JOUR AU - Bogso, Antoine Marie TI - An application of multivariate total positivity to peacocks JO - ESAIM: Probability and Statistics PY - 2014 SP - 514 EP - 540 VL - 18 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2013049/ DO - 10.1051/ps/2013049 LA - en ID - PS_2014__18__514_0 ER -
Bogso, Antoine Marie. An application of multivariate total positivity to peacocks. ESAIM: Probability and Statistics, Volume 18 (2014), pp. 514-540. doi: 10.1051/ps/2013049
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