Multivariate exponential integral approximations: a moment approach Dimitris Bertsimas * Xuan Vinh Doan † Jean Lasserre ‡ January 2007 Abstract We propose a method to approximate a class of exponential multivariate integrals using moment relaxations. Using this approach, both lower and upper bounds of the integrals are obtained and we show that these bound values asymptotically converge to the real value of the integrals when the moment degree r increases. We further demonstrate the method by calculating both hypercubic and order statistic probabilities for multivariate normal distributions. 1 Introduction Multivariate integrals arise in statistic, physics, engineering and finance applications among other areas. For example, these integrals are needed to calculate probabilities over compact sets for multivariate normal random variables. It is therefore important to compute or approximate multivariate integrals. Usual methods include Monte Carlo schemes (see Niederreiter [7] for details) and cubature formulae as shown in e.g. de la Harpe and Pache [2]. However, there are still many open problems currently and research on multivariate integrals is very much active due to its importance as well as its difficulties. For instance, most cubature formulas are restricted to special sets like boxes and simplices, and even in this particular context, determination of orthogonal polynomials used to construct a cubature, is not an easy task. * Boeing Professor of Operations Research, Sloan School of Management, co-director of the Operations Research Center, Massachusetts Institute of Technology, E40-147, Cambridge, MA 02139-4307, dbertsim@mit.edu. † Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, vanxuan@mit.edu. ‡ LAAS, 7 Avenue Du Colonel Roche, 31077 Toulouse C´ edex 4, France, lasserre@laas.fr 1