Penalized nonparametric drift estimation for a multidimensional diffusion process

Abstract

We consider a multi-dimensional diffusion process $\left(\mathbf{X}_{t}\right)_{t\geq0}$ with drift vector $\mathbf{b}$ and diffusion matrix $\Sigma$. This process is observed at $n+1$ discrete times with regular sampling interval $\Delta$. We provide sufficient conditions for the existence and unicity of an invariant density. In a second step, we assume that the process is stationary, and estimate the drift function $\mathbf{b}$ on a compact set $K$ in a nonparametric way. For this purpose, we consider a family of finite dimensional linear subspaces of $L^{2}\left(K\right)$, and compute a collection of drift estimators on every subspace by a penalized least squares approach. We introduce a penalty function and select the best drift estimator. We obtain a bound for the risk of the resulting adaptive estimator. Our method fits for any dimension $d$, but, for safe of clarity, we focus on the case $d=2$. We also provide several examples of two-dimensional diffusions satisfying our assumptions, and realize various simulations. Our results illustrate the theoretical properties of our estimators

Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations*

“…Perhaps more importantly, under some extra, but standard assumptions in non-parametric inference for multidimensional stochastic differential equations (cf. Dalalyan and Reiß (2007) and Schmisser (2013)), we show that our analysis in the one-dimensional case extends to the multidimensional setting as well. To the best of our knowledge, this is a new result in this context.…”

“…Remark 17. Examples of multidimensional stochastic differential equations satisfying assumptions in Definition 7 are given in Section 5.2 in Schmisser (2013).…”

Non-parametric adaptive estimation of the drift for a jump diffusion process

“…3. We consider the vectorial subspaces S m,r generated by the spline functions of degree r (see for instance Schmisser (2013)). In that case D m,r = dim(S m,r ) = 2 m + r. For r ∈ {1, 2, 3} and m ∈ M n (r) = {m, D m,r ≤ D n }, we compute the estimatorsb m,r andb m,r by minimising the contrast functions γ n andγ n on the vectorial subspaces S m,r .…”

Bidimensional random effect estimation in mixed stochastic differential model

“…The nonparametric estimation of the drift function in SDEs with no random effect has been largely investigated in the literature (see e.g. Kutoyants, 2004;Comte et al, 2007;Schmisser, 2013). In the case of SDEs with random effects and a general drift b(x, ϕ 1 , ϕ 2 ), the nonparametric estimation of the function b is open and of interest.…”