New results for drift estimation in inhomogeneous stochastic differential equations

We consider $N$ independent and identically distributed (i.i.d.) stochastic processes $(X_j\left(t\right),t\in [0,T])$, $j\in \{1,\dots ,N\}$, defined by a one-dimensional stochastic differential equation (SDE) with time-dependent drift and diffusion coefficient. In this context, the nonparametric estimation of a general drift function $b(t,x)$ from a continuous observation of the $N$ sample paths on $[0,T]$ has never been investigated. Considering a set $I_{ϵ}=[ϵ,T]\times A$, with $ϵ\ge 0$ and $A\subset R$, we build by a projection method an estimator of $b$ on $I_{ϵ}$. As the function is bivariate, this amounts to estimating a matrix of projection coefficients instead of a vector for univariate functions. We make use of Kronecker products, which simplifies the mathematical treatment of the problem. We study the risk of the estimator and distinguish the case where $ϵ=0$ and the case $ϵ>0$ and $A=[a,b]$ compact. In the latter case, we investigate rates of convergence and prove a lower bound showing that our estimator is minimax. We propose a data-driven choice of the projection space dimension leading to an adaptive estimator. Examples of models and numerical simulation results are proposed. The method is easy to implement and works well, although computationally slower than for the estimation of a univariate function.