Approximating the Laplace transform of the sum of dependent lognormals

Let $(X_1,\dots ,X_n)$ be multivariate normal, with mean vector $\mathit{\mu }$ and covariance matrix $\mathbf{\Sigma }$, and $S_n={\mathrm{e}}^{X_1}+\cdots +{\mathrm{e}}^{X_n}$. The Laplace transform ${\cal L}(\theta)=$\mathbb{E}$$\mathrm{e}$^{-\theta S_n} \propto \int \exp{-h_\theta(\boldsymbol{x})} ,$\mathrm{d}$ \boldsymbol{x}$ is represented as $\tilde{\mathcal{L}}(\theta )I(\theta )$, where $\tilde{\mathcal{L}}(\theta )$ is given in closed-form and $I(\theta )$ is the error factor ($\approx 1$). We obtain $\tilde{\mathcal{L}}(\theta )$ by replacing $h_{\theta }(\mathit{x})$ with a second order Taylor expansion around its minimiser ${\mathit{x}}^{\ast }$. An algorithm for calculating the asymptotic expansion of ${\mathit{x}}^{\ast }$ is presented, and it is shown that $I(\theta )\to 1$ as $\theta \to \mathrm{\infty }$. A variety of numerical methods for evaluating $I(\theta )$ are 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.