Regression Estimation and Prediction in Continuous Time

One of the most important problems in the statistical analysis of stochastic processes is statistical prediction. Many people have worked on that topic but we do not have enough space here to quote them all. However it is interesting to point out that Akaike wrote a paper on prediction in 1969, and that the famous Akaike criterion is based on the prediction error. Since the best probabilistic predictor, i.e. the conditional expectation of the future given the past, is unknown in general, the statistician must start his study by estimating the associated regression problem. In the Markovian case, or if some martingale approximation is available, the final predictor is then obtained by plugging-in the last observations. In such a framework, the choice of a parametric model has some drawbacks. In particular it generates a “mechanic effect” on the predictor which often results in a poor efficiency. The aim of this paper is to present new results on nonparametric regression estimation and prediction in continuous time. Actually, estimation from continuous time observations has received renewed insight in the last few years. This phenomenon may be linked with the various applications that do exist, for example, in medicine, finance or physics, and also to the recent progress in data processing. Moreover, contrary to the classical (linear or non linear) regression problem where the parameter is finite-dimensional, no functional form (except some kind of smoothness conditions) is required or imposed in the nonparametric framework. One thereby avoids the delicate problem of choosing an adequate regression model. For a detailed discussion about nonparametric versus parametric models, we refer to Carbon and Delecroix (1993). Here we focus on the popular kernel method based on the regularization of empirical measures by convolution. This method (jointly introduced by Nadaraya