On MAP estimates and source conditions for drift identification in SDEs

We consider the inverse problem of identifying the drift in an SDE from $n$ observations of its solution at $M+1$ distinct time points. We derive a corresponding MAP estimate, we prove differentiability properties as well as a so-called tangential cone condition for the forward operator, and we review the existing theory for related problems, which under a slightly stronger tangential cone condition would additionally yield convergence rates for the MAP estimate as $n\to \mathrm{\infty }$. Numerical simulations in 1D indicate that such convergence rates indeed hold true.