Importance sampling for efficient parametric simulation
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Monte Carlo simulation, while flexible and widely applicable, has a relatively slow rate of convergence and significant effort has been devoted to improving its efficiency. In this thesis, we focus on the efficient simulation technique of Importance Sampling (IS) in simulation settings where the objective is to improve computational efficiency when solving several instances of an estimation problem and not a single instance only. In the setting of parametric estimation where instances of the estimation problem differ in some model or decision parameters, we consider using the optimal sampling measure at a nominal parameter as the sampling measure at neighboring parameters and analyze the variance of the resulting importance sampling estimator. We show that the resulting estimator can be very effective in estimations in the vicinity of the nominal parameter. We describe an approach to implement this scheme using an approach called the DataBase Monte Carlo (DBMC). Generalizations of DBMC to Importance Sampling and combining Importance Sampling with Stratification are introduced. Next we consider the application of Importance Sampling to a problem in computational finance, namely, portfolio risk evaluation. We use a linear approximation to the portfolio loss, as a function of the underlying risk factors, to obtain a good importance sampling density. We derive the optimal direction for shifting the means of the underlying risk factor. We apply this approach to estimate the probability of large portfolio losses in both one-level and nested simulation settings. In the one-level simulation, it is assumed that the approximation to the loss function is available. In the context of nested simulation, we consider an initial learning phase to "learn" the approximation function using "a regression-like" approach. In cases where the linear approximation is inadequate, we propose an approach that combines Importance Sampling with Stratificiation by subdividing the sampling space into strata and then applying Importance Sampling in each subdivision.
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