Gaussian Volterra processes with power-type kernels. Part I

Abstract

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}}

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具有幂型核的高斯Volterra过程。第一部分

形式的随机过程 \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] ，其中W是标准维纳过程， $\alpha >-\frac{1}{2}$， $\gamma >-1$，和 $\alpha +\beta +\gamma >-\frac{3}{2}$． 证明了过程X是定义良好的连续过程。研究了过程X的增量的方差和界的渐近性质。并证明了过程X满足单点Hölder条件 $\alpha +\beta +\gamma +\frac{3}{2}$ 在点0处，“间隔”Hölder条件达到顺序 $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ 在间隔上 $[{t_{0}},T]$ (哪里 $0<{t_{0}}

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