Additive Regression Model for Continuous Time Processes

The multivariate regression function estimation is an important problem which has been extensively treated for discrete time processes. It is well-known from (11) that the additive regression models bring out a solution to the problem of the curse of dimensionality in nonparametric multivariate regression estimation, which is characterized by a loss in the rate of convergence of the regression function estimator when the dimension of the covariates increases. Additive models allow to reach even univariate rate when these models fit well. For continuous time processes, (2) obtained the optimal rate for the estimator of multivariate regression, which is the same as in the i.i.d. case. He even proved that, for processes with irregular paths, it is possible to reach the parametric rate. This one, called the superoptimal rate, does not depend on the dimension of the variables, but the needed conditions on the processes are very strong. That is the reason why it is relevant to study additive models to bring out a solution to the problem of the curse of dimensionality.