Nonparametric drift estimation for i.i.d. paths of censored stochastic differential equations (SDE)
Résumé
We study nonparametric drift estimation for a diffusion observed under continuous-time right censoring. The latent process X solves dXt = b(Xt) dt + σ(Xt) dWt, while the observer records Yt = min(Xt, Ct) together with the censoring indicator δt = 1{Xt < Ct}, where C is a continuous semimartingale independent of X. Based on N i.i.d. trajectories over a fixed time horizon [0, T], we construct a problem-specific contrast and a projection least-squares estimator of b on approximation spaces. We establish nonasymptotic risk bounds, propose an adaptive dimension selection rule with explicit penalties, and derive convergence rates over Besov balls on compact domains and Sobolev-type classes on non-compact domains. We also introduce a diagnostic procedure to assess the level of censoring and illustrate, through simulations, how censoring mechanisms affect estimation performance.
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