Stochastic analysis for vector-valued generalized grey Brownian motion

IF 0.4 Q4 STATISTICS & PROBABILITY Theory of Probability and Mathematical Statistics Pub Date : 2021-11-17 DOI:10.1090/tpms/1184
W. Bock, M. Grothaus, Karlo S. Orge
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引用次数: 3

Abstract

In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a d d -dimensional Donsker’s delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in d d dimensions.
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向量值广义灰布朗运动的随机分析
本文证明了广义灰布朗运动(ggBm)的标准向量值推广具有独立分量,当且仅当它是分数布朗运动。为了扩展具有独立分量的ggBm,我们引入了一个向量值广义灰布朗运动(vggBm)。相应测度的特征函数被引入为一维情形的特征函数的乘积。我们证明了对于这个测度,Appel系统和广义函数或分布的微积分是可访问的。我们用适当的变换刻画了这些分布,并给出了一个d维Donsker的delta函数作为这种分布的例子。由此,我们证明了vggBm的局部时间和自交局部时间作为分布在某些约束下的存在性,并计算了它们相应的广义期望。最后,我们在d维中求解了一个由vggBm噪声驱动的线性SDE系统。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
22
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