Unobserved expected returns in a diffusive price process: is filtering effective?

We consider a stock price that follows a diffusion process whose drift is mean-reverting and unobservable. We learn about the latent process by filtering the observed log-returns via the Kalman filter. The parameters of the latent process are estimated using two different methodologies. First, we employ the expectation-maximization (EM) algorithm, a maximum likelihood method tailored for such settings. Next, we maximize the likelihood of the innovation process. We assess the estimation performance of both methodologies using simulated data. The surprising result is that even with 30 years of daily data, substantial estimation errors persist across many individual path realizations. We attempt to explain this finding by recognizing that the actual importance of the hidden factor within the observable process is determined by the ratio of their volatilities scaled by the inverse frequency of observations. This scaling factor renders the latent process elusive within the observed time-series.