Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-08-21T07:26:44.886Z Has data issue: false hasContentIssue false
  • English
  • Français

An application of multivariate total positivity to peacocks

Published online by Cambridge University Press:  10 October 2014

Antoine Marie Bogso
Antoine Marie Bogso*
Affiliation:
Universitéde Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, 54506, France CNRS, Institut Elie Cartan de Lorraine, UMR 7502, 54506, Vandoeuvre-lès-Nancy, France. antoine.bogso@univ-lorraine.fr; ambogso@gmail.com
Get access

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.

Keywords

Convex orderpeacockstotal positivity of order 2 (TP2)multivariate total positivity of order 2 (MTP2)markov propertystrong conditional monotonicity
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 514 - 540
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M.Y. An, Log-concave probability distributions: Theory and statistical testing. SSRN (1997) i–29.
Berk, R.H., Some monotonicity properties of symmetric Pólya densities and their exponential families. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42 (1978) 303–307. Google Scholar
D. Baker, C. Donati-Martin and M. Yor, 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.
A.M. Bogso, Étude de peacocks sous des hypothèses de monotonie conditionnelle et de positivité totale. Thèse de l’Université de Lorraine (2012).
A.M. Bogso, C. Profeta and B. Roynette, Some examples of peacocks in a Markovian set-up, in Séminaire de Probabilités, XLIV. Lect. Notes Math. Springer, Berlin (2012) 281–315.
A.M. Bogso, C. Profeta and B. Roynette. Peacocks obtained by normalisation, strong and very strong peacocks, in Séminaire de Probabilités, XLIV. Lect. Notes Math. Springer, Berlin (2012) 317–374.
Baker, D. and Yor, M., A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion. Elect. J. Probab. 14 (2009) 1532–1540. Google Scholar
D. Baker and M. Yor, On martingales with given marginals and the scaling property, in Séminaire de Probabilités XLIII. Lect. Notes Math. Springer, Berlin (2010) 437–439.
Carr, P., Ewald, C.-O. and Xiao, Y., On the qualitative effect of volatility and duration on prices of Asian options. Finance Res. Lett. 5 (2008) 162–171. Google Scholar
Daduna, H. and Szekli, R., A queueing theoretical proof of increasing property of Pólya frequency functions. Statist. Probab. Lett. 26 (1996) 233–242. Google Scholar
Edrei, A., On the generating function of a doubly infinite, totally positive sequence. Trans. Amer. Math. Soc. 74 (1953) 367–383. Google Scholar
Efron, B., Increasing properties of Pólya frequency functions. Ann. Math. Statist. 36 (1965) 272–279. Google Scholar
Émery, M. and Yor, M., A parallel between Brownian bridges and gamma bridges. Publ. Res. Inst. Math. Sci. 40 (2004) 669–688. Google Scholar
W. Feller, Diffusions processes in genetics. Proc. of Second Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley (1951) 227–246.
P.J. Fitzsimmons, J.W. Pitman and M. Yor, Markovian bridges: construction, Palm interpretation and splicing. Seminar on Stochastic Processes (Seattle, WA, 1992). Progr. Probab. Birkhäuser Boston, Boston, MA 33 (1993) 101–134.
Gupta, R.D. and Richards, D. St. P., Multivariate Liouville distributions. J. Multivariate Anal. 23 (1987) 233–256. Google Scholar
K. Hamza and F.C. Klebaner, A family of non-Gaussian martingales with Gaussian marginals. J. Appl. Math. Stoch. Anal. (2007) 92723.
F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011).
Hirsch, F. and Roynette, B., A new proof of Kellerer Theorem. ESAIM: PS 16 (2012) 48–60. Google Scholar
Hirsch, F., Roynette, B. and Yor, M., 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. Google Scholar
Hirsch, F., Roynette, B. and Yor, M., Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets. Expositiones Math. 4 (2010) 299–324. Google Scholar
Hirsch, F., Roynette, B. and Yor, M., 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. Google Scholar
Karlin, S., Total positivity, absorption probabilities and applications, Trans. Amer. Math. Soc. 111 (1964) 33–107. Google Scholar
S. Karlin, Total positivity. Stanford University Press (1967).
Karlin, S. and McGregor, J.L., Coincidence probabilities, Pacific J. Math. 9 (1959) 1141–1165. Google Scholar
Karlin, S. and McGregor, J.L.. Classical diffusion processes and total positivity, J. Math. Anal. Appl. 1 (1960) 163–183. Google Scholar
S. Karlin and H.M. Taylor, A second course in Stochastic processes. Academic Press, New York (1981).
Kellerer, H.G., Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198 (1972) 99–122. Google Scholar
Karlin, S. and Rinot, Y., Classes of orderings of measures and related correlation inequalities I. Multivariate totally positive distributions. J. Multivariate Anal. 10 (1980) 467–498. Google Scholar
Müller, A. and Scarsini, M., Stochastic comparison of random vectors with a common copula. Math. Operat. Res. 26 (2001) 723–740. Google Scholar
G. Pagès, 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.
Rothschild, M. and Stiglitz, J.E., Increasing risk I. A definition. J. Econom. Theory 2 (1970) 225–243. Google Scholar
D. Revuz and M. Yor, Continuous martingales and Brownian motion, vol. 293. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd edition (1999).
T.K. Sarkar, Some lower bounds of reliability. Technical Report, No. 124, Dept. of Operations Research and Statistics, Stanford University (1969).
Schoenberg, I.J., On Pólya frequency functions I. The totally positive functions and their Laplace transforms. J. Analyse Math. 1 (1951) 331–374. Google Scholar
M. Shaked and J.G. Shanthikumar, Stochastic orders and their applications. Probab. Math. Statistics. Academic Press, Boston (1994).