APPROXIMATING THE LAPLACE TRANSFORM OF THE SUM OF DEPENDENT LOGNORMALS

Let (X-1, . . , X-n) be multivariate normal, with mean vector mu and covariance matrix Sigma, and let Sn = e(XI) + . . . + e(Xn). The Laplace transform L(0) = Ee(-theta Sn) alpha integral exp{-h(theta)(x)} dx is represented as (L) over tilde(theta) I (theta), where (L) over tilde(theta) is given in closed form and 1(theta) is the error factor (approximate to 1). We obtain (L) over tilde(theta) by replacing h(theta)(x) with a second-order Taylor expansion around its minimiser x*. An algorithm for calculating the asymptotic expansion of x* is presented, and it is shown that I(theta) -> 1 as theta -> infinity. A variety of numerical methods for evaluating I (theta) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density of S-n) are also given.