Weighted Sub-fractional Brownian Motion Process: Properties and Generalizations

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® \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 $a\in (-1,\mathrm{\infty })$ such that $0\le f(u)\le c{u}^a$ 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.