Maximum likelihood estimation for multiscale Ornstein-Uhlenbeck processes

We study the problem of estimating the parameters of an Ornstein-Uhlenbeck (OU) process that is the coarse-grained limit of a multiscale system of OU processes, given data from the multiscale system. We consider both the averaging and homogenization cases and both drift and diffusion coefficients. By restricting ourselves to the OU system, we are able to substantiall y improve the results in [23, 21] and provide some intuition of what to expect in the general case. In particular, in the homogenisation case we derive optimal rates of sub-sampling, proving the conjecture in [23]