Parametric Estimation from Approximate Data: Non-Gaussian Diffusions

We study the problem of parameters estimation in Indirect Observability contexts, where $X_t \in R^r$ is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process $Y^{\varepsilon}_t$ which is close to $X_t$ in $L^4$ norm. We construct consistent parameter estimators which are smooth functions of the sub-sampled empirical mean and empirical lagged covariance matrices computed from the observable data. We derive explicit optimal sub-sampling schemes specifying the best paired choices of sub-sampling time-step and number of observations. We show that these choices ensure that our parameter estimators reach optimized asymptotic $L^2$-convergence rates, which are constant multiples of the $L^4$ norm $|| Y^{\varepsilon}_t - X_t ||$.