Copulas: a Review and Recent Developments (2007)

0 2 4 6 8 10 120 1 2 3 4 5 60 2 4 6 8 10Daily returns of Korea & Singapore0 1 2 3 4 5 6Standardized residuals of Korea & SingaporeFigure 1: Excesses (left) and standardized residuals excesses (right)Because of space limitations, it was impossible even to comment the di±cultieswith copulas associated to discrete multivariate distributions, the important aspects incopula theory related to stochastic ordering, e.g. MÄuller and Stoyan (2001), graphicaltechniques, e.g. Genest and Boies (2003), classical and new measures of dependence,e.g. Nelsen (1999) and Genest and Plante (2003), etc. A part of these interestingtopics are discussed in Anjos et al. (2004). We would like to mention ¯nally theimportance of survival copulas in life modelling. An excellent overview is given byGeorges et al. (2001), see also Nelsen (2005b).6 AknowledgementsApartofthepresentedmaterialinSection3was,infact,aplanned¯rststepsoftheresearch project related with a visit of the ¯rst author in CIMAT, Guanajuato, Mexicoin the beginning of 2003. As a result, in Kolev et al. (2006) is given an alternativede¯nition of the probability generating function of arbitrary binary vector, wherecomonotonic and countermonotonic discrete copulas play a key role.Special thanks to the organizers of the 8th Symposium of Probability and StochasticProcesses provided in Puebla, Mexico, June 20-25, 2004, where the authors havebeen invited to present their recent results.The authors are thankful to Professor Roger Nelsen for the continuous attentionon our research. We are greatful to the annonimous referee whose critical remarksreduced the errors and improved this overview.The ¯rst and the third authors are partially supported by FAPESP, Grant 03/10105-2 and PROBRAL (CAPES/DAAD), Grant 171-04. The second author thanks for a¯nancial support from CNPq and FAPERJ Grant E-26/170.725/2004.36