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Stochastic emulation techniques represent a specialized surrogate modeling branch that is appropriate for applications for which the relationship between input and output is stochastic in nature. Their objective is to address the stochastic uncertainty sources by directly predicting the output distribution for a given input. An example of such application, and the focus of this contribution, is the estimation of structural response (engineering demand parameter) distribution in seismic risk assessment. In this case, the stochastic uncertainty originates from the aleatoric variability in the seismic hazard description. Note that this is a different uncertainty-source than the potential parametric uncertainty associated with structural characteristics or explanatory variables for the seismic hazard (for example, intensity measures), that are treated as the parametric input in surrogate modeling context. The key challenge in stochastic emulation pertains to addressing heteroscedasticity in the output variability. Relevant approaches to-date for addressing this challenge have focused on scalar outputs. In contrast, this paper focuses on the multi-output stochastic emulation problem and presents a methodology for predicting the output correlation matrix, while fully addressing heteroscedastic characteristics. This is achieved by introducing a Gaussian Process (GP) regression model for approximating the components of the correlation matrix, and coupling this approximation with a correction step to guarantee positive definite properties for the resultant predictions. For obtaining the observation data to inform the GP calibration, different approaches are examined, relying-or-not on the existence of replicated samples for the response output. Such samples require that, for a portion of the training points, simulations are repeated for the same inputs and different descriptions of the stochastic uncertainty. This information can be readily used to obtain observation for the response statistics (correlation or covariance in this instance) to inform the GP development. An alternative approach is to use as observations noisy covariance samples based on the sample deviations from a primitive mean approximation. These different observation variants lead to different GP variants that are compared within a comprehensive case study. A computational framework for integrating the correlation matrix approximation within the stochastic emulation for the marginal distribution approximation of each output component is also discussed, to provide the joint response distribution approximation.