On parameter identification in stochastic differential equations by penalized maximum likelihood

In this paper we present nonparametric estimators for coefficients in stochastic differential equation if the data are described by independent, identically distributed random variables. The problem is formulated as a nonlinear ill-posed operator equation with a deterministic forward operator described by the Fokker-Planck equation. We derive convergence rates of the risk for penalized maximum likelihood estimators with convex penalty terms and for Newton-type methods. The assumptions of our general convergence results are verified for estimation of the drift coefficient. The advantages of log-likelihood compared to quadratic data fidelity terms are demonstrated in Monte-Carlo simulations.