Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes

Statistical inference for Vasicek-type model driven by Hermite processes

Let $Z$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. When $q=1$, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as $q\geq 2$. In this paper, we deal with a Vasicek-type model driven by $Z$, of the form $dX_t = a(b - X_t)dt +dZ_t$. Here, $a > 0$ and $b \in \mathbb{R}$ are considered as unknown drift parameters. We provide estimators for $a$ and $b$ based on continuous-time observations. For all possible valu…

Multifractional Hermite processes: definition and first properties

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener-Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener …

Hermite过程驱动过程的线性乘数非参数估计

Nonparametric estimation of linear multiplier for processes driven by a Hermite process

本文研究了由Hermite过程驱动的随机过程的线性乘数函数的非参数估计问题，分析了估计量在噪声趋于零时的渐近行为。

Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

Let $(Z^{q, H}_t)_{t \in [0, 1]^d}$ denote a $d$-parameter Hermite random field of order $q \geq 1$ and self-similarity parameter $H = (H_1, \ldots, H_d) \in (\frac{1}{2}, 1)^d$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion ($q=1$, $d=1$), fractional Brownian sheet $(q=1, d \geq 2)$, Rosenblatt process ($q=2$, $d=1$) as well as Rosenblatt sheet $(q=2, d \geq 2)$. For any $q \geq 2, d\geq 1$ and $H \in (\frac{1}{2}, 1)^d$ we show in this paper that a proper normalization of the quadratic va…

Non-Central Limit Theorem for Quadratic Functionals of Hermite-Driven Long Memory Moving Average Processes

Let $(Z_t^{(q, H)})_{t \geq 0}$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. Consider the Hermite-driven moving average process $$X_t^{(q, H)} = \int_0^t x(t-u) dZ^{(q, H)}(u), \qquad t \geq 0.$$ In the special case of $x(u) = e^{-θu}, θ> 0$, $X$ is the non-stationary Hermite Ornstein-Uhlenbeck process of order $q$. Under suitable integrability conditions on the kernel $x$, we prove that as $T \to \infty$, the normalized quadratic functional $$G_T^{(q, H)}(t)=\frac{1}{T^{2H_0 - 1}}\int_0^{Tt}\Big(\big(X_s^{(q, H)}\big)^2 - E\Big[\big(X_s^…

对由维纳或赫尔米特过程驱动的均质极限进行统计推断

Statistical Inference for Homogenization Limits Driven by Wiener or Hermite Processes

研究慢-快系统均质极限下的扩散系数和Hurst参数估计，通过维纳或赫尔米特过程驱动的随机微分方程，利用重正化二次变分和Wiener展开证明估计器的一致性及非中心极限定理。