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.
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Statistics, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2013, 47 (1), pp 61-84. 〈10.1080/02331888.2011.591931〉
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Contributeur : Emeline Schmisser <>
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Emeline Schmisser. Penalized nonparametric drift estimation for a multidimensional diffusion process. Statistics, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2013, 47 (1), pp 61-84. 〈10.1080/02331888.2011.591931〉. 〈hal-00358410〉

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