Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions

We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain $D$. We prove that a smooth density exists on $D$ and give upper bounds for this density. Under some additional conditions (mainly dealing with the growth of the coefficients and their derivatives), we formulate upper bounds that are suitable to obtain asymptotic estimates of the density for large values of the state variable ("tail" estimates). These results specify and extend some results by Kusuoka and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1--76], but our approach is substantially different and based on a technique to estimate the Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008) 135--156] and Bally [Integration by parts formula for locally smooth laws and applications to equations with jumps I (2007) The Royal Swedish Academy of Sciences]. This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, that is, a situation where standard techniques for density estimation based on Malliavin calculus do not apply. We establish the existence of a smooth density, for which we give exponential estimates and study the behavior at the origin (the singular point).Comment: Published in at http://dx.doi.org/10.1214/10-AAP717 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The cost of sustainability on optimal portfolio choices

We examine the impact of sustainability criteria, as measured by the KLD scores, on optimal portfolio selection performed on an investment universe containing the equities in the S&P500 index and covering the period between 1993 and 2008. The optimizations are done according to the Markowitz mean-variance approach while sustainability constraints are introduced by eliminating from the investment pool those assets that do not comply to di®erent social responsibility criteria (screening). We compare the two efficient frontiers, i.e. the one without and the one with screening. A spanning test is performed to determine if the differences between the two types of efficient frontier are significant. We introduce a measure of how much an investor has to pay (through loss of return or through additional risk) in order to satisfy given sustainability criteria. The analysis is carried on separately on the three main dimensions of sustainability, namely Environmental, Social and Governance.Sustainability; Optimal portfolio