We consider a diffusion model
dX
t
= b(X
t
)dt + σ(X
t
)dW
t
,X
0
= η,
under conditions ensuring existence, stationarity and geometrical
β
-mixing of the process solution. We assume that we observe a ...sample (
X
k
Δ
)
0≤k≤n+1
. Our aim is to study nonparametric estimators of the drift function b(.), under general conditions. We propose projection estimators based on a least-squares type contrast and, in order to generalize existing results, we want to consider possibly non compactly supported projection bases and possibly non bounded volatility. To that aim, we relate the model with a simpler regression model, then to a more elaborate heteroscedastic model, plus some residual terms. This allows to see the role of heteroscedasticity first and the role of dependency between the variables and to present different probabilistic tools used to face each part of the problem. For each step, we try to see the “price” of each assumption.
This is the developed version of the talk given in August 2018 in Dijon, Journées MAS.
Nous considérons un modèle de diffusion
dX
t
= b(X
t
)dt + σ(X
t
)dW
t
,X
0
= η,
sous des conditions garantissant l’existence, la stationarité et le
β
-mélange géométrique du processus solution. Nous supposons que nous disposons d’observations
(X
kΔ
)
0≤k≤n
. Notre objectif est d’étudier un estimateur nonparamétrique de la fonction
b(.),
sous des hypothèses générales. Nous proposons des estimateurs par projection, basés sur un contraste des moindres carrés. Afin de généraliser les résultats existants, nous voulons des jeux d’hypothèses autorisant des bases de projection à support non compact, ainsi que des fonctions de volatilité non bornées. Ainsi, nous relions le modèle de diffusion à un modèle plus simple de régression, puis à un modèle hétéroscédastique, plus des termes de reste. Cela nous permet de détailler le rôle de l’hétéroscédasticité puis de la dépendance, et de présenter les différents outils probabilistes utilisés pour affronter chaque problème. A chaque étape, nous étudions le prix des hypothèses.
Ceci est la version développée de l’exposé présenté en Août 2018 à Dijon, lors des Journées MAS.
Nonparametric estimation for I.I.D. paths of fractional SDE Comte, Fabienne; Marie, Nicolas
Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems,
10/2021, Letnik:
24, Številka:
3
Journal Article
Odprti dostop
This paper deals with nonparametric estimators of the drift function
b
computed from independent continuous observations, on a compact time interval, of the solution of a stochastic differential ...equation driven by the fractional Brownian motion (fSDE). First, a risk bound is established on a Skorokhod’s integral based least squares oracle
b
^
of
b
. Thanks to the relationship between the solution of the fSDE and its derivative with respect to the initial condition, a risk bound is deduced on a calculable approximation of
b
^
. Another bound is directly established on an estimator of
b
′
for comparison. The consistency and rates of convergence are established for these estimators in the case of the compactly supported trigonometric basis or the
R
-supported Hermite basis.
In this paper, we study nonparametric estimation of the Lévy density for Lévy processes, with and without Brownian component. For this, we consider n discrete time observations with step Δ. The ...asymptotic framework is: n tends to infinity, Δ = Δ n tends to zero while nΔ n tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator of the Lévy density and to provide a bound for the global 𝕃²-risk. Estimators of the drift and of the variance of the Gaussian component are also studied. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.
Nonparametric estimation for i.i.d. Gaussian continuous time moving average models Comte, Fabienne; Genon-Catalot, Valentine
Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems,
04/2021, Letnik:
24, Številka:
1
Journal Article
Odprti dostop
We consider a Gaussian continuous time moving average model
X
(
t
)
=
∫
0
t
a
(
t
-
s
)
d
W
(
s
)
where
W
is a standard Brownian motion and
a
(.) a deterministic function locally square integrable on
...R
+
. Given
N
i.i.d. continuous time observations of
(
X
i
(
t
)
)
t
∈
0
,
T
on 0,
T
, for
i
=
1
,
⋯
,
N
distributed like
(
X
(
t
)
)
t
∈
0
,
T
, we propose nonparametric projection estimators of
a
2
under different sets of assumptions, which authorize or not fractional models. We study the asymptotics in
T
,
N
(depending on the setup) ensuring their consistency, provide their nonparametric rates of convergence on functional regularity spaces. Then, we propose a data-driven method corresponding to each setup, for selecting the dimension of the projection space. The findings are illustrated through a simulation study.
We consider deconvolution from repeated observations with unknown error distribution. Until now, this model has mostly been studied under the additional assumption that the errors are symmetric.
We ...construct an estimator for the non-symmetric error case and study its theoretical properties and practical performance. It is interesting to note that we can improve substantially upon the rates of convergence which have been presented in the literature and, at the same time, dispose of most of the extremely restrictive assumptions which have been imposed so far.
In the present paper, we consider that N diffusion processes X1,…,XN are observed on 0,T, where T is fixed and N grows to infinity. Contrary to most of the recent works, we no longer assume that the ...processes are independent. The dependency is modeled through correlations between the Brownian motions driving the diffusion processes. A nonparametric estimator of the drift function, which does not use the knowledge of the correlation matrix, is proposed and studied. Its integrated mean squared risk is bounded and an adaptive procedure is proposed. Few theoretical tools to handle this kind of dependency are available, and this makes our results new. Numerical experiments show that the procedure works in practice.
This paper studies a classical extension of the Black and Scholes model for option pricing, often known as the Hull and White model. Our specification is that the volatility process is assumed not ...only to be stochastic, but also to have long‐memory features and properties. We study here the implications of this continuous‐time long‐memory model, both for the volatility process itself as well as for the global asset price process. We also compare our model with some discrete time approximations. Then the issue of option pricing is addressed by looking at theoretical formulas and properties of the implicit volatilities as well as statistical inference tractability. Lastly, we provide a few simulation experiments to illustrate our results.
For n independent random variables having the same Hölder continuous density, this paper deals with controls of the Wolverton–Wagner’s estimator MSE and MISE. Then, for a bandwidth hn(β), estimators ...of β are obtained by a Goldenshluger–Lepski type method and a Lacour–Massart–Rivoirard type method. Some numerical experiments are provided for this last method.
We study non parametric drift estimation for an ergodic diffusion process from discrete observations. The drift is estimated on a set A using an approximate regression equation by a least squares ...contrast, minimized over finite dimensional subspaces of L2(A,dx). The novelty is that the set A is non compact and the diffusion coefficient unbounded. Risk bounds of a L2-risk are provided where new variance terms are exhibited. A data-driven selection procedure is proposed where the dimension of the projection space is chosen within a random set contrary to usual selection procedures.
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