# An application of multivariate total positivity to peacocks

ESAIM: Probability and Statistics (2014)

- Volume: 18, page 514-540
- ISSN: 1292-8100

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topBogso, Antoine Marie. "An application of multivariate total positivity to peacocks." ESAIM: Probability and Statistics 18 (2014): 514-540. <http://eudml.org/doc/273648>.

@article{Bogso2014,

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.},

author = {Bogso, Antoine Marie},

journal = {ESAIM: Probability and Statistics},

keywords = {convex order; peacocks; total positivity of order 2 (TP2); multivariate total positivity of order 2 (MTP2); markov property; strong conditional monotonicity; total positivity of order 2 $(TP_\{2\})$; multivariate total positivity of order 2 $(MTP_\{2\})$; Markov property},

language = {eng},

pages = {514-540},

publisher = {EDP-Sciences},

title = {An application of multivariate total positivity to peacocks},

url = {http://eudml.org/doc/273648},

volume = {18},

year = {2014},

}

TY - JOUR

AU - Bogso, Antoine Marie

TI - An application of multivariate total positivity to peacocks

JO - ESAIM: Probability and Statistics

PY - 2014

PB - EDP-Sciences

VL - 18

SP - 514

EP - 540

AB - 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.

LA - eng

KW - convex order; peacocks; total positivity of order 2 (TP2); multivariate total positivity of order 2 (MTP2); markov property; strong conditional monotonicity; total positivity of order 2 $(TP_{2})$; multivariate total positivity of order 2 $(MTP_{2})$; Markov property

UR - http://eudml.org/doc/273648

ER -

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