Multilevel monte carlo for smoothing via transport methods

In this article we consider recursive approximations of the smoothing distribution associated to partially observed stochastic differential equations (SDEs), which are observed discretely in time. Such models appear in a wide variety of applications including econometrics, finance, and engineering. This problem is notoriously challenging, as the smoother is not available analytically and hence requires numerical approximation. This usually consists of applying a time-discretization to the SDE, for instance the Euler method, and then applying a numerical (e.g., Monte Carlo) method to approximate the smoother. This has led to a vast literature on methodology for solving such problems, perhaps the most popular of which is based upon the particle filter (PF), e.g., [A. Doucet and A. Johansen, Handbook of Nonlinear Filtering, Oxford University Press, 2011]. In the context of filtering for this class of problems, it is well known that the particle filter can be improved upon in terms of cost to achieve a given mean squared error (MSE) for estimates. This is in the sense that the computational effort can be reduced to achieve this target MSE by using multilevel methods [M. B. Giles, Oper. Res., 56 (2008), pp. 607–617; M. B. Giles, Acta Numer., 24 (2015), pp. 259–328; S. Heinrich, in Large-Scale Scientific Computing, Springer, New York, 2001] via the multilevel particle filter (MLPF) [A. Gregory, C. Cotter, and S. Reich, SIAM J. Sci. Comp., 38 (2016), pp. A1317–A1338; A. Jasra, K. Kamatani, K. J. Law, and Y. Zhou, SIAM J. Numer. Anal., 55 (2017), pp. 3068–3096; A. Jasra, K. Kamatani, P. Osei, and Y. Zhou, Stat. Comp., 28 (2018), pp. 47–60]. For instance, to obtain a MSE of O(2) for some > 0 when approximating filtering distributions associated with Euler-discretized diffusions with constant diffusion coefficients, the cost of the PF is O(− 3) while the cost of the MLPF is O(− 2 log()2). In this article we consider a new approach to replace the PF, using transport methods in [A. Spantini, D. Bigoni, and Y. Marzouk, Inference via Low-Dimensional Couplings, preprint, arXiv:1703.06131, 2017]. In the context of filtering, one expects that the proposed method improves upon the MLPF by yielding, under assumptions, a MSE of O(2) for a cost of O(− 2). This is established theoretically in an “ideal” example and numerically in numerous examples.