Optimal convergence rates for the invariant density estimation of jump-diffusion processes

Amorino, Chiara; Nualart, Eulàlia

doi:10.1051/ps/2022001

Cited by 3 publications

(2 citation statements)

References 37 publications

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“…with K a kernel function as in (4). The asymptotic behaviour of the estimator proposed in ( 5) is based on the bias-variance decomposition.…”

Section: Construction Estimator and Main Resultsmentioning

confidence: 99%

“…Regarding the issue of nonparametric invariant density estimation, recent contributions in this context include [4] and [16] starting from the observation of diffusion with jumps, [8] and [13] for diffusions driven by a fractional Brownian motion, [15] for interacting particles system, [30] for iid copies of diffusions as in (1). The nonparametric estimation of the invariant density starting from the observation of a stochastic differential equation has recently been studied in [38] and [5] assuming that the continuous record of the process is available.…”

Section: Introductionmentioning

confidence: 99%

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Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime

Amorino¹,

Gloter²

2022

Preprint

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Let (Xt) t≥0 be solution of a one-dimensional stochastic differential equation. Our aim is to study the convergence rate for the estimation of the invariant density in intermediate regime, assuming that a discrete observation of the process (Xt) t∈[0,T ] is available, when T tends to ∞. We find the convergence rates associated to the kernel density estimator we proposed and a condition on the discretization step ∆n which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is, where β is the smoothness of the invariant density. After that, we complement the upper bounds previously found with a lower bound over the set of all the possible estimator, which provides the same convergence rate: it means it is not possible to propose a different estimator which achieves better convergence rates. This is obtained by the two hypothesis method; the most challenging part consists in bounding the Hellinger distance between the laws of the two models. The key point is a Malliavin representation for a score function, which allows us to bound the Hellinger distance through a quantity depending on the Malliavin weight.

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“…with K a kernel function as in (4). The asymptotic behaviour of the estimator proposed in ( 5) is based on the bias-variance decomposition.…”

Section: Construction Estimator and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime

Amorino¹,

Gloter²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

Amorino

Gloter

2023

Statistics

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The problem of estimating a parameter in the drift coefficient is addressed for N discretely observed independent and identically distributed stochastic differential equations (SDEs). This is done considering additional constraints, wherein only public data can be published and used for inference. The concept of local differential privacy (LDP) is formally introduced for a system of stochastic differential equations. The objective is to estimate the drift parameter by proposing a contrast function based on a pseudo-likelihood approach. A suitably scaled Laplace noise is incorporated to meet the privacy requirements. Our key findings encompass the derivation of explicit conditions tied to the privacy level. Under these conditions, we establish the consistency and asymptotic normality of the associated estimator. Notably, the convergence rate is intricately linked to the privacy level, and in some situations may be completely different from the case where privacy constraints are ignored. Our results hold true as the discretization step approaches zero and the number of processes N tends to infinity.

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Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Hölder classes

Amorino,

Gloter

2024

Scandinavian J Statistics

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We study the problem of the nonparametric estimation for the density of the stationary distribution of a ‐dimensional stochastic differential equation . From the continuous observation of the sampling path on , we study the estimation of as goes to infinity. For , we characterize the minimax rate for the ‐risk in pointwise estimation over a class of anisotropic Hölder functions with regularity . For , our finding is that, having ordered the smoothness such that , the minimax rate depends on whether or . In the first case, this rate is , and in the second case, it is , where is an explicit exponent dependent on the dimension and , the harmonic mean of smoothness over the directions after excluding and , the smallest ones. We also demonstrate that kernel‐based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both integrated and pointwise risk. In the two‐dimensional case, we show that kernel density estimators achieve the rate , which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results.

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Optimal convergence rates for the invariant density estimation of jump-diffusion processes

Cited by 3 publications

References 37 publications

Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime

Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity

Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Hölder classes

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