From regression function to diffusion drift estimation in nonparametric setting

We consider a diffusion model dXt = b(Xt)dt + σ(Xt)dWt,X0 = η, under conditions ensuring existence, stationarity and geometrical β-mixing of the process solution. We assume that we observe a sample (XkΔ)0≤k≤n+1. Our aim is to study nonparametric estimators of the drift function b(.), under general conditions. We propose projection estimators based on a least-squares type contrast and, in order to generalize existing results, we want to consider possibly non compactly supported projection bases and possibly non bounded volatility. To that aim, we relate the model with a simpler regression model, then to a more elaborate heteroscedastic model, plus some residual terms. This allows to see the role of heteroscedasticity first and the role of dependency between the variables and to present different probabilistic tools used to face each part of the problem. For each step, we try to see the “price” of each assumption. This is the developed version of the talk given in August 2018 in Dijon, Journées MAS.