Hi,

Here are the most intersting reference I could find

I post them because I think other people might be intereted in this issue,
so here are the articles titles :

Asmussen, Rojas-Nandayapa - Sums of Dependent lognormal Random Variables, Asymptotics and Simulation

Vanduffel, Chen, Dhaene, Goovaerts, Henrard - Optimal Approximations for Risk Measures of Sums of Lognormals based on Conditional Expectations

Li - A Novel Accurate Approximation Method of Lognormal Sum Random Variables

Gao, Xu, Ye- Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

Mehta, Molisch, Wu, Zhang - Approximating the Sum of Correlated Lognormal or Lognormal-Rice Random Variables

Fu, Madan, Wang - Pricing Continuous Asian Options, A Comparison of Monte Carlo and Laplace Transform Inversion Methods

Vecer - New Pricing of Asian Options

And a few other articles not directly applicable to this problem but interesting on their own :

Eden, Viens - General Upper and Lower Tail Estimates using Malliavin Calculus and Stein's Equations

Barndorf-Nielsen, Kluppelberg - Tail Exactness of Multivariate Saddlepoint Approximations

Have a nice day