{"title":"Weighted Sub-fractional Brownian Motion Process: Properties and Generalizations","authors":"Ramirez-Gonzalez Jose Hermenegildo, Sun Ying","doi":"arxiv-2409.04798","DOIUrl":null,"url":null,"abstract":"In this paper, we present several path properties, simulations, inferences,\nand generalizations of the weighted sub-fractional Brownian motion. A primary\nfocus is on the derivation of the covariance function $R_{f,b}(s,t)$ for the\nweighted sub-fractional Brownian motion, defined as: \\begin{equation*}\nR_{f,b}(s,t) = \\frac{1}{1-b} \\int_{0}^{s \\wedge t} f(r) \\left[(s-r)^{b} +\n(t-r)^{b} - (t+s-2r)^{b}\\right] dr, \\end{equation*} where $f:\\mathbb{R}_{+} \\to\n\\mathbb{R}_{+}$ is a measurable function and $b\\in [0,1)\\cup(1,2]$. This\ncovariance function $R_{f,b}(s,t)$ is used to define the centered Gaussian\nprocess $\\zeta_{t,f,b}$, which is the weighted sub-fractional Brownian motion.\nFurthermore, if there is a positive constant $c$ and $a \\in (-1,\\infty)$ such\nthat $0 \\leq f(u) \\leq c u^{a}$ on $[0,T]$ for some $T>0$. Then, for $b \\in\n(0,1)$, $\\zeta_{t,f,b}$ exhibits infinite variation and zero quadratic\nvariation, making it a non-semi-martingale. On the other hand, for $b \\in\n(1,2]$, $\\zeta_{t,f,b}$ is a continuous process of finite variation and thus a\nsemi-martingale and for $b=0$ the process $\\zeta_{t,f,0}$ is a square\nintegrable continuous martingale. We also provide inferential studies using\nmaximum likelihood estimation and perform simulations comparing various\nnumerical methods for their efficiency in computing the finite-dimensional\ndistributions of $\\zeta_{t,f,b}$. Additionally, we extend the weighted\nsub-fractional Brownian motion to $\\mathbb{R}^d$ by defining new covariance\nstructures for measurable, bounded sets in $\\mathbb{R}^d$. Finally, we define a\nstochastic integral with respect to $\\zeta_{t,f,b}$ and introduce both the\nweighted sub-fractional Ornstein-Uhlenbeck process and the geometric weighted\nsub-fractional Brownian motion.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04798","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we present several path properties, simulations, inferences,
and generalizations of the weighted sub-fractional Brownian motion. A primary
focus is on the derivation of the covariance function $R_{f,b}(s,t)$ for the
weighted sub-fractional Brownian motion, defined as: \begin{equation*}
R_{f,b}(s,t) = \frac{1}{1-b} \int_{0}^{s \wedge t} f(r) \left[(s-r)^{b} +
(t-r)^{b} - (t+s-2r)^{b}\right] dr, \end{equation*} where $f:\mathbb{R}_{+} \to
\mathbb{R}_{+}$ is a measurable function and $b\in [0,1)\cup(1,2]$. This
covariance function $R_{f,b}(s,t)$ is used to define the centered Gaussian
process $\zeta_{t,f,b}$, which is the weighted sub-fractional Brownian motion.
Furthermore, if there is a positive constant $c$ and $a \in (-1,\infty)$ such
that $0 \leq f(u) \leq c u^{a}$ on $[0,T]$ for some $T>0$. Then, for $b \in
(0,1)$, $\zeta_{t,f,b}$ exhibits infinite variation and zero quadratic
variation, making it a non-semi-martingale. On the other hand, for $b \in
(1,2]$, $\zeta_{t,f,b}$ is a continuous process of finite variation and thus a
semi-martingale and for $b=0$ the process $\zeta_{t,f,0}$ is a square
integrable continuous martingale. We also provide inferential studies using
maximum likelihood estimation and perform simulations comparing various
numerical methods for their efficiency in computing the finite-dimensional
distributions of $\zeta_{t,f,b}$. Additionally, we extend the weighted
sub-fractional Brownian motion to $\mathbb{R}^d$ by defining new covariance
structures for measurable, bounded sets in $\mathbb{R}^d$. Finally, we define a
stochastic integral with respect to $\zeta_{t,f,b}$ and introduce both the
weighted sub-fractional Ornstein-Uhlenbeck process and the geometric weighted
sub-fractional Brownian motion.