The fractional multivariate normal tempered stable process (Q714607)

A fractional tempered stable subordinator \((T_H(t))_{t \geq 0}\) is a fractional Lévy process, defined in this paper as \[ T_H(t) = \int_0^t K_{H,\alpha}(t,u)dT(u), \] where \(K_{H,\alpha}\) is the Volterra kernel and \((T(t))_{t \geq 0}\) denotes a tempered stable subordinator with parameters \((\alpha, k, \theta)\), \(\alpha \in (0,2)\). (Detailed definitions are given in the paper.) Then, \(T_H\) exhibits long-range dependence when \(H \in \left(\frac{1}{\alpha}, \frac{1}{\alpha}+\frac12\right)\) and short-range dependence when \(H \in \left(\frac{1}{\alpha}-\frac12, \frac{1}{\alpha}\right]\). Let \(B_{H_1} = (B_{H_1,1}, B_{H_1, 2}, \dotsc, B_{H_1, N})\) be a multivariate fractional Brownian motion with Hurst index \(H_1 \in (0, 1]\) and with a given covariance matrix \(\Sigma t^{2H_1}\), \(t \geq 0\). Also, let \(T_{H_2}\) be a fractional tempered stable subordinator defined above, independent of \(B_{H_1}\). Then, a fractional multivariate normal tempered stable process \(X\), the main interest of the paper, is defined by the author as \[ X(t) = (T_{H_2}(t))^{2H_1} \beta + B_{H_1}(T_{H_2}(t)), \quad \beta \in \mathbb{R}^N . \] The paper describes the characteristic and moment generating functions and the covariance matrix of this process. In addition, it uses a series representation of \(T_{H_2}\) from \textit{C. Houdré} and \textit{R. Kawai} [Stochastic Processes Appl. 116, No. 8, 1161--1184 (2006; Zbl 1102.60036)], for a numerical method to simulate sample paths of the above defined process. (This reference is a major source of the paper.)