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.
Mots-clés : 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 = {http://archive.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 - http://archive.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, Tome 18 (2014), pp. 514-540. doi : 10.1051/ps/2013049. http://archive.numdam.org/articles/10.1051/ps/2013049/
[1] M.Y. An, Log-concave probability distributions: Theory and statistical testing. SSRN (1997) i-29.
[2] , Some monotonicity properties of symmetric Pólya densities and their exponential families. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42 (1978) 303-307. | MR | Zbl
[3] , and , A sequence of Albin type continuous martingales, with Brownian marginals and scaling, in Séminaire de Probabilités XLIII. Lect. Notes Math. Springer, Berlin (2011) 441-449. | MR | Zbl
[4] , Étude de peacocks sous des hypothèses de monotonie conditionnelle et de positivité totale. Thèse de l'Université de Lorraine (2012).
[5] , and , Some examples of peacocks in a Markovian set-up, in Séminaire de Probabilités, XLIV. Lect. Notes Math. Springer, Berlin (2012) 281-315. | MR | Zbl
[6] , and . Peacocks obtained by normalisation, strong and very strong peacocks, in Séminaire de Probabilités, XLIV. Lect. Notes Math. Springer, Berlin (2012) 317-374. | MR | Zbl
[7] and , A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion. Elect. J. Probab. 14 (2009) 1532-1540. | MR | Zbl
[8] and , On martingales with given marginals and the scaling property, in Séminaire de Probabilités XLIII. Lect. Notes Math. Springer, Berlin (2010) 437-439. | MR | Zbl
[9] , and , On the qualitative effect of volatility and duration on prices of Asian options. Finance Res. Lett. 5 (2008) 162-171.
[10] and , A queueing theoretical proof of increasing property of Pólya frequency functions. Statist. Probab. Lett. 26 (1996) 233-242. | MR | Zbl
[11] , On the generating function of a doubly infinite, totally positive sequence. Trans. Amer. Math. Soc. 74 (1953) 367-383. | MR | Zbl
[12] , Increasing properties of Pólya frequency functions. Ann. Math. Statist. 36 (1965) 272-279. | MR | Zbl
[13] and , A parallel between Brownian bridges and gamma bridges. Publ. Res. Inst. Math. Sci. 40 (2004) 669-688. | MR | Zbl
[14] , Diffusions processes in genetics. Proc. of Second Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley (1951) 227-246. | MR | Zbl
[15] , and , Markovian bridges: construction, Palm interpretation and splicing. Seminar on Stochastic Processes (Seattle, WA, 1992). Progr. Probab. Birkhäuser Boston, Boston, MA 33 (1993) 101-134. | MR | Zbl
[16] and , Multivariate Liouville distributions. J. Multivariate Anal. 23 (1987) 233-256. | MR | Zbl
[17] and , A family of non-Gaussian martingales with Gaussian marginals. J. Appl. Math. Stoch. Anal. (2007) 92723. | MR | Zbl
[18] , , and , Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011). | Zbl
[19] and , A new proof of Kellerer Theorem. ESAIM: PS 16 (2012) 48-60. | Numdam | MR | Zbl
[20] , and , From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order. J. Math. Soc. Japan 63 (2011) 887-917. | MR | Zbl
[21] , and , Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets. Expositiones Math. 4 (2010) 299-324. | MR | Zbl
[22] , and , Applying Itô's motto: look at the infinite dimensional picture by constructing sheets to obtain processes increasing in the convex order. Periodica Math. Hungarica 61 (2010) 195-211. | MR | Zbl
[23] , Total positivity, absorption probabilities and applications, Trans. Amer. Math. Soc. 111 (1964) 33-107. | MR | Zbl
[24] , Total positivity. Stanford University Press (1967). | Zbl
[25] and , Coincidence probabilities, Pacific J. Math. 9 (1959) 1141-1165. | MR | Zbl
[26] and . Classical diffusion processes and total positivity, J. Math. Anal. Appl. 1 (1960) 163-183. | MR | Zbl
[27] and , A second course in Stochastic processes. Academic Press, New York (1981). | MR | Zbl
[28] , Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198 (1972) 99-122. | MR | Zbl
[29] and , Classes of orderings of measures and related correlation inequalities I. Multivariate totally positive distributions. J. Multivariate Anal. 10 (1980) 467-498. | MR | Zbl
[30] and , Stochastic comparison of random vectors with a common copula. Math. Operat. Res. 26 (2001) 723-740. | MR | Zbl
[31] , Functional co-monotony of processes with an application to peacocks and barrier options, in Séminaire de Probabilités XLV. Lect. Notes Math. Springer (2013) 365-400. | MR | Zbl
[32] and , Increasing risk I. A definition. J. Econom. Theory 2 (1970) 225-243. | MR
[33] and , Continuous martingales and Brownian motion, vol. 293. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd edition (1999). | MR | Zbl
[34] , Some lower bounds of reliability. Technical Report, No. 124, Dept. of Operations Research and Statistics, Stanford University (1969). | MR
[35] , On Pólya frequency functions I. The totally positive functions and their Laplace transforms. J. Analyse Math. 1 (1951) 331-374. | MR | Zbl
[36] and , Stochastic orders and their applications. Probab. Math. Statistics. Academic Press, Boston (1994). | MR | Zbl
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