Gaussian Volterra processes: asymptotic growth and statistical estimation

遍历 Volterra Ornstein-Uhlenbeck 过程中的最大似然估计

Maximum likelihood estimation in the ergodic Volterra Ornstein-Uhlenbeck process

本文研究遍历 Volterra Ornstein-Uhlenbeck 过程的漂移参数推断，分别在连续、高频和低频观测下建立 MLE 与矩估计的渐近性质。

小噪声渐近下粗糙过程的漂移估计：QMLE方法

Drift estimation for rough processes under small noise asymptotic : QMLE approach

本文研究了在小噪声渐近条件下，通过QMLE方法对具有奇异核的随机Volterra过程的漂移函数参数进行估计，证明了在网格大小趋于零时，路径重建误差随h^1/2衰减，从而得到一个有效的估计量。

Convergence of long-memory discrete $k$-th order Volterra processes

We obtain limit theorems for a class of nonlinear discrete-time processes $X(n)$ called the $k$-th order Volterra processes of order $k$. These are moving average $k$-th order polynomial forms: \[ X(n)=\sum_{0<i_1,\ldots,i_k<\infty}a(i_1,\ldots,i_k)ε_{n-i_1}\ldotsε_{n-i_k}, \] where $\{ε_i\}$ is i.i.d.\ with $\mathbb{E} ε_i=0$, $\mathbb{E} ε_i^2=1$, where $a(\cdot)$ is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for $X(n)$ to be well-defined in $L^2(Ω)$, and focus on central and non-central limit theorems. We show that normalized partia…

Volterra bootstrap: Resampling higher-order statistics for strictly stationary univariate time series

We are concerned with nonparametric hypothesis testing of time series functionals. It is known that the popular autoregressive sieve bootstrap is, in general, not valid for statistics whose (asymptotic) distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or a nonlinear one. Inspired by nonlinear system theory we circumvent this non-validity by introducing a higher-order bootstrap scheme based on the Volterra series representation of the process. In order to estimate coefficients of such a representation efficiently, we rely …

Stochastic Generalized Lotka-Volterra Model with An Application to Learning Microbial Community Structures

Inferring microbial community structure based on temporal metagenomics data is an important goal in microbiome studies. The deterministic generalized Lotka-Volterra differential (GLV) equations have been used to model the dynamics of microbial data. However, these approaches fail to take random environmental fluctuations into account, which may negatively impact the estimates. We propose a new stochastic GLV (SGLV) differential equation model, where the random perturbations of Brownian motion in the model can naturally account for the external environmental effects on the microbial community. …

Simulation of volatility modulated Volterra processes using hyperbolic stochastic partial differential equations

We propose a finite difference scheme to simulate solutions to a certain type of hyperbolic stochastic partial differential equation (HSPDE). These solutions can in turn estimate so called volatility modulated Volterra (VMV) processes and Lévy semistationary (LSS) processes, which is a class of processes that have been employed to model turbulence, tumor growth and electricity forward and spot prices. We will see that our finite difference scheme converges to the solution of the HSPDE as we take finer and finer partitions for our finite difference scheme in both time and space. Finally, we dem…