GLOTER Arnaud

The subject of the thesis is parametric and non-parametric estimation in jump process models. The thesis is composed of 3 parts which regroup 4 works. The first part, which is composed of two chapters, deals with the estimation of drift and volatility parameters by contrast methods from discrete observations, with the main objective of minimizing the conditions on the observation step, so that it can for example go arbitrarily slowly towards 0. The second part of the thesis concerns asymptotic developments, and bias correction, for the estimation of the integrated volatility. The third part of the thesis, concerns the adaptive estimation of the stationary measure for jump processes.

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.

We study the problem of the non-parametric estimation for the density π of the stationary distribution of a stochastic two-dimensional damping Hamiltonian system (Z_t) t∈[0,T ] = (X_t, Y_t) t∈[0,T ]. From the continuous observation of the sampling path on [0, T ], we study the rate of estimation for π(x_0 , y_0) as T → ∞. We show that kernel based estimators can achieve the rate T^{−v} for some explicit exponent v ∈ (0, 1/2). One finding is that the rate of estimation depends on the smoothness of π and is completely different with the rate appearing in the standard i.i.d. setting or in the case of two-dimensional non degenerate diffusion processes. Especially, this rate depends also on y 0. Moreover, we obtain a minimax lower bound on the L 2-risk for pointwise estimation, with the same rate T^{−v}, up to log(T) terms.

No summary available.

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 ∆n = sup i=0,.,n−1 (t n i+1 − t n i) → 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 ∆n → 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∆ k n → 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, Lévy-driven SDE, thresholding methods.

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 ∆n = sup i=0,.,n−1 (t n i+1 − t n i) → 0. We introduce an estimator of θ, based on a contrast function, which is efficient without requiring any conditions on the rate at which ∆n → 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n∆ 3 n → 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∆ k n → 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. Lévy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.

The problem of integrated volatility estimation for the solution X of a stochastic differential equation with Lévy-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). Lévy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data.

We consider the solution X = (Xt) t≥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≤t≤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 Hölder 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 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, Lévy driven SDE.

We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter θ_1 in a non-degenerate diffusion coefficient and a parameter θ_2 in the drift term. The second component has a drift term parameterized by θ_3 and no diffusion term. Asymptotic normality is proved in three different situations for an adaptive estimator for θ_3 with some initial estimators for (θ_1 , θ_2), an adaptive one-step estimator for (θ_1 , θ_2 , θ_3) with some initial estimators for them, and a joint quasi-maximum likelihood estimator for (θ_1 , θ_2 , θ_3) without any initial estimator. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for θ_1 is smaller than the standard one based only on the first component. The convergence of the estimators for θ_3 is much faster than the other parameters. The resulting asymptotic variance is smaller than that of an estimator only using the increments of the second component.

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al.

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 ∆n = sup i=0,.,n−1 (t n i+1 − t n i) → 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 ∆n → 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∆ k n → 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, Lévy-driven SDE, thresholding methods.

This paper is concerned with parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2) and does not require any integrability assumptions on the process. The main limit theorems are derived thanks to a control in total variation distance between the law of the normalized process, in small time, and the α-stable distribution. This method is an alternative to the non Gaussian quasi-likelihood estimation method proposed by Masuda [20] where the Blumenthal-Getoor index α is restricted to belong to the interval [1, 2).

The problem of integrated volatility estimation for the solution X of a stochastic differential equation with Lévy-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). Lévy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data.

Considering a class of stochastic differential equations driven by a locally stable process, we address the joint parametric estimation, based on high frequency observations of the process on a fixed time interval , of the drift coefficient, the scale coefficient and the jump activity of the process. This work extends [4] where the jump activity was assumed to be known and also [3] where the LAN property and the estimation of the three parameters are performed for a translated stable process. We propose an estimation method and show that the asymptotic properties of the estimators depend crucially on the form of the scale coefficient. If the scale coefficient is multiplicative: a(x, σ) = σa(x), the rate of convergence of our estimators is non diagonal and the asymptotic variance in the joint estimation of the scale coefficient and the jump activity is the inverse of the information matrix obtained in [3]. In the non multiplicative case, the results are better and we obtain a faster diagonal rate of convergence with a different asymptotic variance. In both cases, the estimation method is illustrated by numerical simulations showing that our estimators are rather easy to implement. MSC 2010 subject classifications: Primary 60G51, 60G52, 60J75, 62F12. secondary 60H07, 60F05 .

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 ∆n = sup i=0,.,n−1 (t n i+1 − t n i) → 0. We introduce an estimator of θ, based on a contrast function, which is efficient without requiring any conditions on the rate at which ∆n → 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n∆ 3 n → 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∆ k n → 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. Lévy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.

In this article we approximate the invariant distribution ν of an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps, particularly suitable in cases where the driving Lévy process is a Compound Poisson. This scheme is similar to those introduced by Lamberton and Pagès in [LP02] for a Brownian diffusion and extended by Panloup in [Pan08b] to the Jump Diffusion with Lévy jumps. We obtain a non-asymptotic Gaussian concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along a appropriate test functions f such that f − ν(f) is is a coboundary of the infinitesimal generator.

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ)T and we study the sensitivity of the density with respect to this parameter. This extends the results of [E.

This paper is concerned with parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2) and does not require any integrability assumptions on the process. The main limit theorems are derived thanks to a control in total variation distance between the law of the normalized process, in small time, and the α-stable distribution. This method is an alternative to the non Gaussian quasi-likelihood estimation method proposed by Masuda [20] where the Blumenthal-Getoor index α is restricted to belong to the interval [1, 2).

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by an α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ) T and we study the sensitivity of the density with respect to this parameter. This extends the results of [5] which was restricted to the index α ∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process. MSC2010: 60G51. 60G52. 60H07. 60H20. 60H10. 60J75.

This work focuses on the Local Asymptotic Mixed Normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a pure jump Lévy process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of [5] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in [6].

It is well known that the strong error approximation in the space of continuous paths equipped with the supremum norm between a diffusion process, with smooth coefficients, and its Euler approximation with step 1/n is O(n−1/2) and that the weak error estimation between the marginal laws at the terminal time T is O(n−1). An analysis of the weak trajectorial error has been developed by Alfonsi, Jourdain and Kohatsu-Higa [Ann. Appl. Probab. 24 (2014) 1049–1080], through the study of the p-Wasserstein distance between the two processes. For a one-dimensional diffusion, they obtained an intermediate rate for the pathwise Wasserstein distance of order n−2/3+ε. Using the Komlós, Major and Tusnády construction, we improve this bound assuming that the diffusion coefficient is linear and we obtain a rate of order logn/n.

It is well known that the strong error approximation, in the space of continuous paths equipped with the supremum norm, between a diffusion process, with smooth coefficients, and its Euler approximation with step 1/n is O(n −1/2) and that the weak error estimation between the marginal laws, at the terminal time T , is O(n −1). An analysis of the weak trajectorial error has been developed by Alfonsi, Jourdain and Kohatsu-Higa [1], through the study of the p−Wasserstein distance between the two processes. For a one-dimensional diffusion, they obtained an intermediate rate for the pathwise Wasserstein distance of order n −2/3+ε. Using the Komlós , Major and Tusnády construction, we improve this bound, assuming that the diffusion coefficient is linear, and we obtain a rate of order log n/n. MSC 2010. 65C30, 60H35.

The problem of drift estimation for the solution $X$ of a stochastic differential equation with L\'evy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density these conditions reduce to $n\Delta_n^{3-\eps}\to 0,$ where $n$ is the number of observations and $\Delta_n$ is the maximal sampling step. This result relaxes the condition $n\Delta_n^2 \to 0$ usually required for joint estimation of drift and diffusion coefficient for SDE's with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part $X^c$ in the likelihood function. In order to construct the drift estimator we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to $X^c$. Convergence results of independent interest are proved for these nonparametric estimators. Finally, we illustrate the behavior of our drift estimator for a number of popular L\'evy--driven models from finance.

We consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. In [13], the evolution of the distance between the two processes, in local time scale and up to their first hitting time is shown to satisfy a stochastic differential equation with jumps. The jumps of this S.D.E. are naturally driven by the excursion process of one of the two skew Brownian motions. In this article, we show that the description of the distance of the two processes after this first hitting time may be studied using the self similarity induced by the previous S.D.E. More precisely, we show that the distance between the two processes in local time scale may be viewed as the unique continuous markovian self-similar extension of the process described in [13]. This permits us to compute the law of the distance of the two skew Brownian motions at any time in the local time scale, when both original skew Brownian motions start from zero. As a by product, we manage to study the markovian dependence on the skewness parameter and answer an open question formulated initially by C. Burdzy and Z.Q. Chen in [6].

No summary available.

We consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. In [13], the evolution of the distance between the two processes, in local time scale and up to their first hitting time is shown to satisfy a stochastic differential equation with jumps. The jumps of this S.D.E. are naturally driven by the excursion process of one of the two skew Brownian motions. In this article, we show that the description of the distance of the two processes after this first hitting time may be studied using the self similarity induced by the previous S.D.E. More precisely, we show that the distance between the two processes in local time scale may be viewed as the unique continuous markovian self-similar extension of the process described in [13]. This permits us to compute the law of the distance of the two skew Brownian motions at any time in the local time scale, when both original skew Brownian motions start from zero. As a by product, we manage to study the markovian dependence on the skewness parameter and answer an open question formulated initially by C. Burdzy and Z.Q. Chen in [6].

No summary available.

During the last decades, the development of technological means and particularly computer science has allowed the emergence of numerical methods for the approximation of Stochastic Differential Equations (SDE) as well as for the estimation of their parameters. This thesis deals with these two aspects and is more specifically interested in the efficiency of these methods. The first part will be devoted to the approximation of SDEs by numerical schemes while the second part deals with the estimation of parameters. In the first part, we study approximation schemes for EDSs. We assume that these schemes are defined on a time grid of size $n$. We will say that the scheme $X^n$ converges weakly to the diffusion $X$ with order $h in mathbb{N}$ if for all $T>0$, $vert mathbb{E}[f(X_T)-f(X_T^n)] vertleqslant C_f /n^h$. Until now, except in some particular cases (Euler and Ninomiya Victoir schemes), the research on the subject imposes that $C_f$ depends on the infinite norm of $f$ but also on its derivatives. In other words $C_f =C sum_{green alpha green leqslant q} Green partial_{alpha} f Green_{ infty}$. Our goal is to show that if the scheme converges weakly with order $h$ for such $C_f$, then, under assumptions of nondegeneracy and regularity of the coefficients, we can obtain the same result with $C_f=C Green f Green_{infty}$. Thus, we prove that it is possible to estimate $mathbb{E}[f(X_T)]$ for $f$ measurable and bounded. We then say that the scheme converges in total variation to the diffusion with order $h$. We also prove that it is possible to approximate the density of $X_T$ and its derivatives by that $X_T^n$. In order to obtain this result, we will use an adaptive Malliavin method based on the random variables used in the scheme. The interest of our approach lies in the fact that we do not treat the case of a particular scheme. Thus our result applies to both Euler ($h=1$) and Ninomiya Victoir ($h=2$) schemes but also to a generic set of schemes. Moreover the random variables used in the scheme do not have imposed probability laws but belong to a set of laws which leads to consider our result as a principle of invariance. We will also illustrate this result in the case of a third order scheme for one-dimensional EDSs. The second part of this thesis deals with the estimation of the parameters of a DHS. Here, we will consider the particular case of the Maximum Likelihood Estimator (MLE) of the parameters that appear in the Wishart matrix model. This process is the multi-dimensional version of the Cox Ingersoll Ross process (CIR) and has the particularity of the presence of the square root function in the diffusion coefficient. Thus this model allows to generalize the Heston model to the case of a local covariance. In this thesis we construct the MLE of the Wishart parameters. We also give the convergence speed and the limit law for the ergodic case as well as for some non-ergodic cases. In order to prove these convergences, we will use various methods, in this case: ergodic theorems, time change methods, or the study of the joint Laplace transform of the Wishart and its mean. Moreover, in this last study, we extend the domain of definition of this joint transform.

We consider a one dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. In that purpose, we adapt the method developed in the ballistic case by Comets et al (2014) and Falconnet, Loukianova and Matias (2014). Using a supplementary assumption due to the specificity of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphazis the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike in the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.

We study the problem of the efficient estimation of the jumps for stochastic processes. We assume that the stochastic jump process $(X_t)_{t \in [0,1]}$ is observed discretely, with a sampling step of size $1/n$. In the spirit of Hajek's convolution theorem, we show some lower bounds for the estimation error of the sequence of the jumps $(\Delta X_{T_k})_k$. As an intermediate result, we prove a LAMN property, with rate $\sqrt{n}$, when the marks of the underlying jump component are deterministic. We deduce then a convolution theorem, with an explicit asymptotic minimal variance, in the case where the marks of the jump component are random. To prove that this lower bound is optimal, we show that a threshold estimator of the sequence of jumps $(\Delta X_{T_k})_k$ based on the discrete observations, reaches the minimal variance of the previous convolution theorem.

We prove the Local Asymptotic Mixed Normality property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a pure jump Lévy process. The process is observed on the fixed time interval [0,1] and the parameter appears in the drift coefficient only. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof of this result contains a sharp study of the asymptotic behavior, in small time, of the transition probability density of the process and of its logarithm derivative.

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.

This thesis deals with the parametric estimation of the drift and diffusion coefficients of a diffusion process when we observe a functional of the trajectory, and not the trajectory itself. The first part of the thesis is devoted to the case where we observe the integral of the trajectory on consecutive time intervals. We are interested in the case where these time intervals are of fixed length and in the case where their length tends to 0. We show explicit contrasts in these two cases which lead to asymptotically Gaussian estimators, easy to implement in practice. The second part is devoted to stochastic volatility models. We consider a two-dimensional process, of which we observe only the first coordinate. This one has for diffusion coefficient the hidden diffusion whose unknown parameters are to be estimated. We construct explicit estimators of all the parameters of the hidden diffusion and determine their convergence speeds and asymptotic laws. Throughout the paper, we illustrate our results with numerical simulations on models commonly used in finance.