Gaussian Processes with Volterra Kernels

The stochastic process of the form \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$ on the entire interval $[0,T]$.

“…Proposition 2. Let the process X admit representation (4), where α, β and γ satisfy relations (2). Then for t 0 > 0 the asymptotic behavior of var(X…”

Section: Asymptotic Properties Of Incremental Variancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gaussian Volterra processes with power-type kernels. Part I

Mishura¹,

Shklyar²

2022

“…First, such kernels are very particular, therefore, it is natural to consider their generalizations and investigate the properties of such generalized GVp's. This was, in particular, done in the papers [16,12,10,15]. Second, the kernels (1), ( 2) are comparatively complicated.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial approach to the calculation of projection coefficients for the simplest Gaussian-Volterra process

Bodnarchuk,

Mishura

2024

Self Cite

“…Such approach is very interesting, but the less facts we know about the kernel of such a representation, the less useful properties of the corresponding process can be established. In this connection, in the paper [9] we considered a kernel of the form…”

Section: Introductionmentioning

confidence: 99%

Gaussian Volterra processes with power-type kernels. Part II

Mishura¹,

Shklyar²

2022

In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three parameters gives us a freedom to operate with the processes and we get a wider application possibilities. On the other hand, it leads to the need to apply rather subtle methods, depending on the intervals where the parameters fall. Integration with respect to the processes under consideration is defined, and it is found for which parameters the processes are differentiable. Finally, the Volterra representation is inverted, that is, the representation of the underlying Wiener process via Gaussian Volterra process is found. Therefore, it is shown that for any indices for which Gaussian Volterra process is defined, it generates the same flow of sigma-fields as the underlying Wiener process – the property that has been used many times when considering a fractional Brownian motion.