On variance reduction of mean-CVaR Monte Carlo estimators

<Para ID="Par1">We formulate an objective as a convex combination of expectation and risk, measured by the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathrm{CVaR }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="normal">CVaR</mi> </math> </EquationSource> </InlineEquation> risk measure. The poor performance of standard Monte Carlo estimators applied on functions of this form is discussed and a variance reduction scheme based on importance sampling is proposed. We provide analytical solution for random variables based on normal distribution and outline the way for the other distributions, either by analytical computation or by sampling. Our results are applied in the framework of stochastic dual dynamic programming algorithm. Computational results which validate the previous analysis are given. Copyright Springer-Verlag Berlin Heidelberg 2015