In this paper we investigate the approximations for the distribution function of a sum S of lognormal random variables. These approximations are obtained by considering the conditional expectation E[S | Λ] of S with respect to a conditioning random variable Λ.The choice of Λ is crucial in order to obtain accurate approximations. The different alternatives for Λ that have been proposed in the literature to date are 'global' in the sense that Λ is chosen such that the entire distribution of the approximation E[S | Λ] is 'close' to the corresponding distribution of the original sum S.In an actuarial or a financial context one is often only interested in a particular tail of the distribution of S. Therefore in this paper we propose approximations E[S | Λ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S | Λ] is an accurate approximation for the corresponding tail of the distribution of S. Numerical illustrations reveal that local optimal choices for Λ can improve the quality of the approximations in the relevant tail significantly.We also explore the asymptotic properties of the approximations E[S | Λ] and investigate links with results from [S. Asmussen, Rojas-Nandayapa, Sums of dependent lognormal random variables: Asymptotics and simulation, Stochastic Series at Department of Mathematical Sciences, University of Aarhus, Research Report number 469, 2005]. Finally, we briefly address the sub-optimality of Asian options from the point of view of risk averse decision makers with a fixed investment horizon.

Nanoparticles with different core@shell structures (NaGdF4:Yb/Tm@NaLnF4 with Ln = lanthanide) were successfully synthesized for luminescence upconversion from 980 nm to 800 nm. The upconversion luminescence of the core@shell nanoparticles (NaGdF4:Yb/Tm@NaGdF4) has a much stronger infrared emission centered at 800 nm than that of the core-only nanoparticles (NaGdF4:Yb/Tm). This is not only due to the removal of surface defects of the core nanoparticles by the shell, which results in the decrease of the luminescence quenching, but also due to the breaking of the crystal field symmetry around the Tm3+ ion. By replacing the NaGdF4 shell with an NaYF4, the emission could further be enhanced. We suggest that the breaking of the crystal field symmetry plays also in this case a key role for further enhancement of upconversion emission. The importance of this asymmetry was further revealed by investigating the energy transfer between Er3+ ion in the shell and the Tm3+ ion of the core nanoparticle.