Gerber and Chopin combine SMC with RQMC to accelerate convergence. They apply RQMC as in the array-RQMC method discussed below, for which convergence rate theory remains thin despite impressive empirical performance. Their proof of o(N −1/2) convergence rate is a remarkable contribution. Array-RQMC simulates an array of N dependent realizations of a Markov chain, using RQMC so the distribution of the N states has low discrepancy D t with respect to its theoretical distribution, at any step t. If x t = Γ t (x t−1 , u t), where x t−1 ∼ U([0, 1)) and u t ∼ U([0, 1) d), array-RQMC approximates the (+ d)-dimensional integral E[D t ], using N RQMC points. It matches each state to a point whose first coordinates are near that point, and uses the next d coordinates for u t. The matching step is crucial. If = 1, just sort the states and points by their (increasing) first coordinate. If > 1, one can map the states x t ∈ [0, 1) → [0, 1), then proceed as for = 1. Gerber and Chopin do exactly this, using a Hilbert curve mapping. But the map [0, 1) → [0, 1) is not essential. E.g., if = 2, one can sort the states and the points in N −1/2 groups of size N −1/2 by their first coordinate, then sort each group by the second coordinate. This extends to > 2. With this multivariate sort, L'Ecuyer et al. (2009) observed an O(N −2) variance when pricing an Asian option with array-RQMC. I wonder what rate the Hilbert curve map can achieve for this example. When X ⊆ R instead of [0, 1) , Gerber and Chopin map x t → [0, 1) using a logistic transformation. The choice of transformation and its parameters may have a significant impact on the overall convergence. For the multivariate sort, no transformation is needed, it works directly in R. The Hilbert sort could also be adapted to work directly in R .

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Pierre l'Ecuyer

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Comment on "Sequential Quasi-Monte Carlo Sampling"