arXiv subjects

Chiara Amorino

Publications and source records attributed to Chiara Amorino.

28 records · Page 2Linked to original sources

arXiv2023

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

We study the problem of the nonparametric estimation for the density $π$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$. From the continuous observation of the sampling path on $[0, T]$, we study the estimation of $π(x)$ as $T$ goes to infinity. For $d\ge2$, we characterize the minimax rate for the $\mathbf{L}^2$-risk in pointwise estimation over a class of anisotropic Hölder functions $π$ with regularity $β= (β_1, ... , β_d)$. For $d \ge 3$, our finding is that, having ordered the smoothness such that $β_1 \le ... \le β_d$, the minimax rate depends on whether $β_2 < β_3$ or $β_2 = β_3$. In the first case, this rate is $(\frac{\log T}{T})^γ$, and in the second case, it is $(\frac{1}{T})^γ$, where $γ$ is an explicit exponent dependent on the dimension and $\barβ_3$, the harmonic mean of smoothness over the $d$ directions after excluding $β_1$ and $β_2$, the smallest ones. We also demonstrate that kernel-based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both $L^2$ integrated and pointwise risk. In the two-dimensional case, we show that kernel density estimators achieve the rate $\frac{\log T}{T}$, which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results.

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arXiv2023

Parameter estimation of discretely observed interacting particle systems

In this paper, we consider the problem of joint parameter estimation for drift and diffusion coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval $[0, T]$, we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($Δ_n$) and the number of particles ($N$) satisfy $Δ_n \rightarrow 0$ and $N \rightarrow \infty$, and asymptotically normal when additionally the condition $Δ_n N \rightarrow 0$ holds.

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arXiv2022

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

We aim at estimating in a non-parametric way the density $π$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$, for $d \ge 2$, from the discrete observations of a finite sample $X_{t_0}$, ... , $X_{t_n}$ with $0= t_0 < t_1 < ... < t_n =: T_n$. We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. Such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder class, associated to $n$ iid random variables $X_1, ..., X_n$. After that we focus on the asynchronous case, in which each component can be observed at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non synchronicity of the data introduces additional bias terms in the study of the estimator.

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arXiv2022

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

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 a long time horizon which remained an open question until now. First, we propose to estimate the volatility coefficient. For that, we introduce a truncation function in our estimation procedure that allows us to take into account the jumps of the process and estimate the volatility function on a linear subspace of L2(A) where A is a compact interval of R. We obtain a bound for the empirical risk of the volatility estimator, ensuring its consistency, and then we study an adaptive estimator w.r.t. the regularity. Then, we define an estimator of a sum between the volatility and the jump coefficient modified with the conditional expectation of the intensity of the jumps. We also establish a bound for the empirical risk for the non-adaptive estimators of this sum, the convergence rate up to the regularity of the true function, and an oracle inequality for the final adaptive estimator.Finally, we give a methodology to recover the jump function in some applications. We conduct a simulation study to measure our estimators' accuracy in practice and discuss the possibility of recovering the jump function from our estimation procedure.

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arXiv2022

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

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 $\frac{1}{T}$, 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 $\frac{\log T}{T}$ 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 $\{\frac{1}{T},\frac{\log T}{T}\}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.

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arXiv2021

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

We study the problem of the non-parametric estimation for the density of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt) , when the dimension d is bigger than 3. From the continuous observation of the sampling path on [0, T ] we show that, under anisotropic Holder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular , 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 Holder smoothness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Holder 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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arXiv2020

Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function

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 at the instants (t n i)i=0,...,n with $Δ$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. 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 $Δ$n $\rightarrow$ 0, assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see [13],[28]) 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$Δ$ k n $\rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [17] in the case of continuous processes. Efficient drift estimation, efficient volatility estimation,ergodic properties, high frequency data, L{é}vy-driven SDE, thresholding methods.

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arXiv2020

Unbiased truncated quadratic variation for volatility estimation in jump diffusion processes

The problem of integrated volatility estimation for the solution X of a stochastic differential equation with L{é}vy-type jumps is considered under discrete high-frequency observations in both short and long 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 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 to have the original truncated realized volatility well-performed (see [22], [13]). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple ($α$, $β$). L{é}vy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data.

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arXiv2020

Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes

We consider the solution X = (Xt) t$\ge$0 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 X T = (Xt) 0$\le$t$\le$T is available. In the case without jumps, Reiss and Dalalyan (2007) and Strauch (2018) have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H{ö}lder 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 $\ge$ 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 Goldensh-luger and Lepski (2011) method which leads us to an adaptive non-parametric kernel estimator of the stationary density $μ$ of the jump diffusion X. Adaptive bandwidth selection, anisotropic density estimation, ergodic diffusion with jumps, L{é}vy driven SDE

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arXiv2019

Contrast function estimation for the drift parameter of ergodic jump diffusion process

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $θ$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $Δ$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an estimator of $θ$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $Δ$n $\rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$Δ$ 3 n $\rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $θ$ is feasible under the condition that n$Δ$ k n $\rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{é}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.

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