2024/09/07 by Ramirez-Gonzalez Jose Hermenegildo, Ying Sun, Hermenegildo, Ramirez-Gonzalez Jose +1
Economics, Econometrics and Finance · #2020 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2409.04798
openalex publication_date 2024/09/07 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28
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 Rf,b(s,t) for the weighted sub-fractional Brownian motion, defined as: Rf,b(s,t) = (1)/(1-b) ∫0s \wedge t f(r) [(s-r)b + (t-r)b - (t+s-2r)b] dr, where f:ℝ+ → ℝ+ is a measurable function and b∈ [0,1)∪(1,2]. This covariance function Rf,b(s,t) is used to define the centered Gaussian process ζt,f,b, which is the weighted sub-fractional Brownian motion. Furthermore, if there is a positive constant c and a ∈ (-1,∞) such that 0 ≤ f(u) ≤ c ua on [0,T] for some T>0. Then, for b ∈ (0,1), ζt,f,b exhibits infinite variation and zero quadratic variation, making it a non-semi-martingale. On the other hand, for b ∈ (1,2], ζt,f,b is a continuous process of finite variation and thus a semi-martingale and for b=0 the process ζt,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 ζt,f,b. Additionally, we extend the weighted sub-fractional Brownian motion to ℝd by defining new covariance structures for measurable, bounded sets in ℝd. Finally, we define a stochastic integral with respect to ζt,f,b and introduce both the weighted sub-fractional Ornstein-Uhlenbeck process and the geometric weighted sub-fractional Brownian motion.