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Robust mean-covariance solutions for stochastic optimization (RV of 2004/04/TM)

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The author provides a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. She proves that, for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program. She first proves a general projection property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance robust objective. This allows the use of known univariate results in the multidimensional setting, and the addition of new results in this direction. In particular, for a general class of objective functions (so called one or two-point support functions), she reduces the robust objective to a deterministic optimization problem in one variable. Finally, she adapts a result from Geoffrion [19] to reduce the main problem to a parametric quadratic program. In particular, her results are true for increasing concave utilities with convex or concave-convex derivatives. She also provides closed form solutions for special quantile-based criteria, such as probability and option-like objectives and value at risk. She investigates applications of her results in portfolio management and multi-product pricing. Finally, she investigates extensions of these results for the case of non-negative and decision dependent returns.

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