Weighted Sub-fractional Brownian Motion Process: Properties and Generalizations

TL;DR

This paper introduces a generalized weighted sub-fractional Brownian motion, exploring its properties, covariance structure, and extensions, along with inference methods, simulations, and new related stochastic processes.

Contribution

It derives the covariance function for weighted sub-fractional Brownian motion, analyzes its path properties, extends it to higher dimensions, and defines related stochastic processes.

Findings

For b in (0,1), the process has infinite variation and zero quadratic variation.

For b in (1,2], the process is of finite variation and semi-martingale.

Simulation studies compare numerical methods for finite-dimensional distributions.

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.