Treffer: On the estimation of the diffusion coefficient for multi-dimensional diffusion processes

The authors consider an estimation problem for the solution \(X\) of a stochastic differential equation of the form \[ dX_ t=b(t,X)dt+a(\vartheta,t,X_ t)dW_ t,\quad{\mathcal L}(X_ 0)=v, \] where the diffusion coefficient \(a\) is a known function of the parameter \(\vartheta\), belonging to a compact set, and the drift \(b\) is a non- anticipative functional. \(X\) is observed at \(n\) distinct times \(t(n,i)\), \(i=1,\dots,n\), in the interval [0,1] chosen such that \(\mu_ n=n^{- 1}\sum_{1\leq i\leq n}{\mathcal E}_{t(n,i)}\) weakly converges to a measure \(\mu\) on [0,1]. It is shown that a consistent sequence \(\widehat\vartheta_ n\) of estimators of \(\vartheta\) can be constructed based on the minimization of suitable constructs and on local asymptotic mixed normality, i.e.: \(\sqrt n(\widehat\vartheta_ n-\vartheta)\) converges in law to a conditional centered Gaussian variable, conditioned on the paths of \(X\). Further, optimality in a certain sense is established under some smoothness assumptions.