论文标题
高斯工艺与Volterra内核
Gaussian processes with Volterra kernels
论文作者
论文摘要
我们研究Volterra流程$ x_t = \ int_0^t k（t，s）dw_s $，其中$ w $是标准的wiener过程，内核具有$ k（t，s）= a（s）= a（s）\ int_s^t b（u）c（u）c（u）c（u）c（u）c（u）c（u）c（u）d $。该表格将沃尔斯特指数$ h> 1/2 $的分数布朗运动（FBM）概括为伏特拉内核。我们建立了$ x $的平滑度属性，包括连续性和持有属性。碰巧的是，其持有人的平滑度接近FBM的众所周知的持有人的平滑度，但更糟。我们与FBM进行比较，以获取任何平滑定理。然后，我们通过$ x $在l^1 [0，t] $创建一个超声波对的情况下，通过$ x $调查了$ w $的反表示问题，即存在$ h \ in l^1 [0，t] $，因此$ c * h = 1 $。它是FBM各自特性的自然扩展，它通过基础维纳过程产生相同的过滤。由于所考虑的高斯过程的反相反表示是基于超声对的属性，因此我们提供了几个众所周知和新的Sonine对示例。 关键词：高斯过程，Volterra过程，超声层，连续性，持有属性，反表示。
We study Volterra processes $X_t = \int_0^t K(t,s) dW_s$, where $W$ is a standard Wiener process, and the kernel has the form $K(t,s) = a(s) \int_s^t b(u) c(u-s) du$. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index $H>1/2$. We establish smoothness properties of $X$, including continuity and Holder property. It happens that its Holder smoothness is close to well-known Holder smoothness of fBm but is a bit worse. We give a comparison with fBm for any smoothness theorem. Then we investigate the problem of inverse representation of $W$ via $X$ in the case where $c\in L^1[0,T]$ creates a Sonine pair, i.e. there exists $h\in L^1[0,T]$ such that $c * h = 1$. It is a natural extension of the respective property of fBm that generates the same filtration with the underlying Wiener process. Since the inverse representation of the Gaussian processes under consideration are based on the properties of Sonine pairs, we provide several examples of Sonine pairs, both well-known and new. Key words: Gaussian process, Volterra process, Sonine pair, continuity, Holder property, inverse representation.
