Efficient exponential tilting with applications

To minimize the variance of Monte Carlo estimators, we develop a novel exponential embedding technique that extends the classical concept of sufficient statistics in importance sampling. Our method demonstrates bounded relative error and logarithmic efficiency when applied to normal and gamma distributions, especially in rare event scenarios. To illustrate this innovative technique, we address the problem of credit risk measurement in portfolios and present an efficient simulation algorithm to estimate the likelihood of significant portfolio losses, leveraging multi-factor models with a normal mixture copula. Finally, supported by comprehensive simulation studies, our approach offers a more effective and efficient way to simulate moderately rare events.