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Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes Amorino, Chiara ; in Journal of Statistical Planning and Inference (in press) We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of ... [more ▼] We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of observations is available. In the case without jumps, Reiss and Dalalyan [7] and Strauch [24] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H ̈older smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d ≥ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [11] which leads us to an adaptive non-parametric kernel estimator of the stationary density μ of the jump diffusion X. [less ▲] Detailed reference viewed: 25 (1 UL)Optimal convergence rates for the invariant density estimation of jump-diffusion processes Amorino, Chiara ; E-print/Working paper (2021) We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of jump diffusion processes whose ... [more ▼] We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate 1/T, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for d = 1 and is equal to log T/T for d = 2. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates { 1/T , log T/T } in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case. [less ▲] Detailed reference viewed: 18 (2 UL)Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes. Amorino, Chiara E-print/Working paper (2020) We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt)0≤t≤T , when the dimension d is ... [more ▼] We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt)0≤t≤T , when the dimension d is such that d ≥ 3. From the continuous observa- tion of the sampling path on [0, T ] we show that, under anisotropic H ̈older smoothness constraints, kernel based estimators can achieve fast convergence rates. In particu- lar, they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic H ̈older smooth- ness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic H ̈older smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose. [less ▲] Detailed reference viewed: 13 (2 UL)On the nonparametric inference of coefficients of self-exciting jump-diffusion Amorino, Chiara ; ; et al E-print/Working paper (2020) In this paper, we consider a one-dimensional diffusion process with jumps driven by a Hawkes process. We are interested in the estimations of the volatility function and of the jump function from discrete ... [more ▼] In this paper, we consider a one-dimensional diffusion process with jumps driven by a Hawkes process. We are interested in the estimations of the volatility function and of the jump function from discrete high-frequency observations in long time horizon. We first propose to estimate the volatility coefficient. For that, we introduce in our estimation procedure a truncation function that allows to take into account the jumps of the process and we estimate the volatility function on a linear subspace of L 2 (A) where A is a compact interval of R. We obtain a bound for the empirical risk of the volatility estimator and establish an oracle inequality for the adaptive estimator to measure the performance of the procedure. Then, we propose an estimator of a sum between the volatility and the jump coefficient modified with the conditional expectation of the intensity of the jumps. The idea behind this is to recover the jump function. We also establish a bound for the empirical risk for the non-adaptive estimator of this sum and an oracle inequality for the final adaptive estimator. We conduct a simulation study to measure the accuracy of our estimators in practice and we discuss the possibility of recovering the jump function from our estimation procedure. [less ▲] Detailed reference viewed: 17 (1 UL)Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function Amorino, Chiara ; in Statistical Inference for Stochastic Processes (2020) In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on μ and volatility coefficient depends on σ, two unknown parameters. We suppose that the process is ... [more ▼] In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on μ and volatility coefficient depends on σ, two unknown parameters. We suppose that the process is discretely observed. We introduce an estimator of θ := (μ, σ), based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which the discretisation step goes to 0, assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see [15],[36]) or where only the estimation of the drift parameter was considered (see [2]). In general situations, our contrast function is not explicit and in practise one has to resort to some approximation. We propose explicit approximations of the contrast function, such that the estimation of θ is feasible under the condition that n∆n^k → 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [24] in the case of continuous processes. [less ▲] Detailed reference viewed: 25 (3 UL)Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes Amorino, Chiara ; in Stochastic Processes and Their Applications (2020) The problem of integrated volatility estimation for an Ito semimartingale is considered under discrete high-frequency observations in short time horizon. We provide an asymptotic expansion for the ... [more ▼] The problem of integrated volatility estimation for an Ito semimartingale is considered under discrete high-frequency observations in short time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results to have the original truncated realized volatility well-performed the condition β> 1 /2 (2− α) on β (that is such that (1/ n)^β is the threshold of the truncated quadratic variation) and on the degree of jump activity α was needed (see Mancini, 2011; Jacod, 2008). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple (α, β). [less ▲] Detailed reference viewed: 17 (0 UL)Contrast function estimation for the drift parameter of ergodic jump diffusion process Amorino, Chiara ; in Scandinavian Journal of Statistics (2020) In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter. We suppose that the process is discretely observed. We introduce an estimator ... [more ▼] In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter. We suppose that the process is discretely observed. We introduce an estimator based on a contrast function, which is efficient without requiring any conditions on the rate at which the step discretization goes to zero, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition on the step discretization was needed and where the process was supposed to have summable jumps. In general situations, our contrast function is not explicit and one has to resort to some approximation. In the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of the drift parameter is feasible. This extends the results obtained by Kessler in the case of continuous processes. [less ▲] Detailed reference viewed: 18 (3 UL) |
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