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We present a method for computing dense stereo correspondences in calibrated monocular video by iteratively and stochastically sampling match quality values in the disparity search space. Most existing methods exhaustively compute local correspondence quality before searching for a globally optimal solution. Instead, we iteratively refine a correspondence estimate by perturbing it with random noise and formulating an influence at each sample based on the perturbation and its effect on correspondence match quality. Local influence is aggregated to recover consistent trends in match quality caused by the piecewise-continuous structure of the scene. Correspondence estimates for a given frame pair are seeded with the estimates from the previous frame pair, allowing convergence to occur across multiple frame pairs.