  In the following article we consider the numerical approximation of the
non-linear filter in continuous-time, where the observations and signal follow
diffusion processes. Given access to high-frequency, but discrete-time
observations, we resort to a first order time discretization of the non-linear
filter, followed by an Euler discretization of the signal dynamics. In order to
approximate the associated discretized non-linear filter, one can use a
particle filter (PF). Under assumptions, this can achieve a mean square error
of $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$ arbitrary, such that the
associated cost is $\mathcal{O}(\epsilon^{-4})$. We prove, under assumptions,
that the multilevel particle filter (MLPF) of Jasra et al (2017) can achieve a
mean square error of $\mathcal{O}(\epsilon^2)$, for cost
$\mathcal{O}(\epsilon^{-3})$. This is supported by numerical simulations in
several examples.
