  Consider the problem of learning the drift coefficient of a stochastic
differential equation from a sample path. In this paper, we assume that the
drift is parametrized by a high dimensional vector. We address the question of
how long the system needs to be observed in order to learn this vector of
parameters. We prove a general lower bound on this time complexity by using a
characterization of mutual information as time integral of conditional
variance, due to Kadota, Zakai, and Ziv. This general lower bound is applied to
specific classes of linear and non-linear stochastic differential equations. In
the linear case, the problem under consideration is the one of learning a
matrix of interaction coefficients. We evaluate our lower bound for ensembles
of sparse and dense random matrices. The resulting estimates match the
qualitative behavior of upper bounds achieved by computationally efficient
procedures.
