In this article, we consider a jump diffusion process (Xt) t≥0 , with drift function b, diffusion coefficient σ and jump coefficient ξ 2 . This process is observed at discrete times t = 0, ∆, . . . , n∆. The sampling interval ∆ tends to 0 and n∆ tends to infinity. We assume that (Xt) t≥0 is ergodic, strictly stationary and exponentially β-mixing. We use a penalized leastsquare approach to compute adaptive estimators of the functions σ 2 + ξ 2 and σ 2 . We provide bounds for the risks of the two estimators. RésuméNous observons une diffusion à sauts (Xt) t≥0 à des instants discrets t = 0, ∆, . . . , n∆. Le temps d'observation n∆ tend vers l'infini et le pas d'observation ∆ tend vers 0). Nous supposons que le processus (Xt) t≥0 est ergodique, stationnaire et exponentiellement β-mélangeant. Nous construisons des estimateurs adaptatifs des fonctions σ 2 + ξ 2 et σ 2 , où σ 2 est le coefficient de diffusions et ξ 2 le coefficient de sauts, grâce à une méthode de moindres carrés pénalisés. Nous majorons le risque de ces estimateurs de manière non asymptotique. ModelWe consider the stochastic differential equation (1). We assume that the following assumptions are fulfilled:

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

International audienceAbstract We consider a diffusion process \left(X_{t}\right)_{t\geq0}, with drift b(x) and diffusion coefficient \sigma(x). At discrete times t_{k}=k\delta for k from 1 to M, we observe noisy data of the sample path, Y_{k\delta}=X_{k\delta}+\varepsilon_{k}. The random variables \left(\varepsilon_{k}\right) are i.i.d, centred and independent of \left(X_{t}\right). The process \left(X_{t}\right)_{t\geq0} is assumed to be strictly stationary, \beta-mixing and ergodic. In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that \Delta=p\delta is small whereas M\delta is large. Then, the diffusion coefficient \sigma^{2} is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diffusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators

In this article, we consider a jump diffusion process (Xt) t≥0 observed at discrete times t = 0, ∆, . . . , n∆. The sampling interval ∆ tends to 0 and n∆ tends to infinity. We assume that (Xt) t≥0 is ergodic, strictly stationary and exponentially β-mixing. We use a penalized least-square approach to compute two adaptive estimators of the drift function b. We provide bounds for the risks of the two estimators.A 1. The functions b, σ and ξ are Lipschitz.A 2.1. The function σ is bounded from below and above:4. The Lévy measure ν satisfies:Under Assumption A1, the stochastic differential equation (1) admits a unique strong solution. According to Masuda (2007), under Assumptions A1 and A2, the process (X t ) admits a unique invariant probability ̟ and satisfies the ergodic theorem: for any measurable function g such that´|g(This distribution has moments of order 4. Moreover, Masuda (2007) also ensures that under these assumptions, the process (X t ) is exponentially β-mixing. Furthermore, if there exist two constants c and n 0 such that, for any x ∈ R, ξ 2 (x) ≥ c(1 + |x|) −n0 , then Ishikawa and Kunita (2006) ensure that a smooth transition density exists.A 3. 1. The stationary measure ̟ admits a density π which is bounded from below and above on the compact interval A:2. The process (X t ) t≥0 is stationary (η ∼ ̟(dx) = π(x)dx).The first part of this assumption is automatically satisfied if ξ = 0 (that is if (X t ) t≥0 is a diffusion process). The following proposition is very useful for the proofs. It is derived from Result 11. Proposition 1.Under Assumptions A1-A3, for any p ≥ 1, there exists a constant c(p) such that, if´R z 2p ν(dz) < ∞: