  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 require numerical approximation. This usually consists
by 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 lead 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. [9]. 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 in the sense that
the computational effort can be reduced to achieve this target MSE, by using
multilevel (ML) methods [12, 13, 18], via the multilevel particle filter (MLPF)
[16, 20, 21]. For instance, to obtain a MSE of $\mathcal{O}(\epsilon^2)$ for
some $\epsilon > 0$ when approximating filtering distributions associated with
Euler-discretized diffusions with constant diffusion coefficients, the cost of
the PF is $\mathcal{O}(\epsilon^{-3})$ while the cost of the MLPF is
$\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In this article we consider a new
approach to replace the particle filter, using transport methods in [27]. In
the context of filtering, one expects that the proposed method improves upon
the MLPF by yielding, under assumptions, a MSE of $\mathcal{O}(\epsilon^2)$ for
a cost of $\mathcal{O}(\epsilon^{-2})$. This is established theoretically in an
"ideal" example and numerically in numerous examples.
