  In this paper, we present a new antithetic multilevel Monte Carlo (MLMC)
method for the estimation of expectations with respect to laws of diffusion
processes that can be elliptic or hypo-elliptic. In particular, we consider the
case where one has to resort to time discretization of the diffusion and
numerical simulation of such schemes. Motivated by recent developments, we
introduce a new MLMC estimator of expectations, which does not require
simulation of intractable L\'evy areas but has a weak error of order 2 and
achieves the optimal computational complexity. We then show how this approach
can be used in the context of the filtering problem associated to partially
observed diffusions with discrete time observations. We illustrate with
numerical simulations that our new approaches provide efficiency gains for
several problems relative to some existing methods.
