Non parametric estimation of the diffusion coefficients of a diffusion with jumps

Schmisser, Emeline

doi:10.1016/j.spa.2019.03.003

Cited by 16 publications

(19 citation statements)

References 28 publications

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“…We underline that Lemma 7 in [30] has been proved for a noisy diffusion. However, the same reasoning applies for a jump diffusion (see the proof of Theorem 13 in [31]) and for our framework as well, as it is based on a projection argument and on algebraic computations which still hold true. From (41) and the fourth point of Assumption 4 we get…”

Section: Proof Of Theoremmentioning

confidence: 80%

“…The literature for the diffusion with jumps from a pure centred Lévy process is large. For example one can refer to [29] and [31].…”

Section: Motivation and State Of The Artmentioning

confidence: 99%

“…Secondly, we want to recover coefficient a. It is important to note that, as presented in [31], in classical jump-diffusion framework (where a Lévy process is used instead of the Hawkes process for M = 1) it is possible to obtain an estimator for the function σ 2 + a 2 by considering the quadratic increments (without truncation) of the process. This is no longer the case here, due to the form of the intensity function of the Hawkes process.…”

Section: Motivation and State Of The Artmentioning

confidence: 99%

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On the nonparametric inference of coefficients of self-exciting jump-diffusion

Amorino¹,

Dion²,

Gloter³

et al. 2020

Preprint

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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.

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Section: Proof Of Theoremmentioning

confidence: 80%

“…The literature for the diffusion with jumps from a pure centred Lévy process is large. For example one can refer to [29] and [31].…”

Section: Motivation and State Of The Artmentioning

confidence: 99%

Section: Motivation and State Of The Artmentioning

confidence: 99%

See 1 more Smart Citation

On the nonparametric inference of coefficients of self-exciting jump-diffusion

Amorino¹,

Dion²,

Gloter³

et al. 2020

Preprint

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show abstract

“…In [13], for example, the authors estimate in a non-parametric way the drift function of a diffusion with jumps driven by a Hawkes process while in [3] the estimation of the integrated volatility is considered. Schmisser investigates, in [31], the non parametric adaptive estimation of the coefficients of a jumps diffusion process and together with Funke she also investigates, in [17], the non parametric adaptive estimation of the drift of an integrated jump diffusion process. Closer to the purpose of this work, in [4] and [2] the convergence rate for the pointwise estimation of the invariant density associated to (1) is considered.…”

Section: Introductionmentioning

confidence: 99%

Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes

Amorino

2021

Electron. J. Statist.

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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 observation of the sampling path on [0, T ] we show that, under anisotropic Hölder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular, they are as fast as the ones found by Dalalyan and Reiss [11] for the estimation of the invariant density in the case without jumps under isotropic Hölder smoothness constraints. Moreover, they are faster than the ones found by Strauch [32] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Hölder 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.

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“…Beyond these works, to our best knowledge, the literature concerning non-parametric estimation of diffusion processes with jumps is not wide. One of the few examples is given by Funke and Schmisser: in [27] they investigate the non parametric adaptive estimation of the drift of an integrated jump diffusion process, while in [40], Schmisser deals with the non-parametric adaptive estimation of the coefficients of a jumps diffusion process. To name other examples, in [23] the authors estimate in a non-parametric way the drift of a diffusion with jumps driven by a Hawkes process, while in [3] the volatility and the jump coefficients are considered.…”

mentioning

confidence: 99%