MIXED GAUSSIAN PROCESSES: A FILTERING APPROACH

• 成果类型: Article
• 署名作者: Cai, Chunhao; Chigansky, Pavel; Kleptsyna, Marina
• 署名单位: Nankai University; Hebrew University of Jerusalem; Le Mans Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1041
• 发表日期: 2016
• 页码: 3032-3075
• 关键词: fractional brownian-motion; EQUIVALENCE

摘要

This paper presents a new approach to the analysis of mixed processes X-t = B-t + G(t), t is an element of [0, T] where B-t is a Brownian motion and G(t) is an independent centered Gaussian process. We obtain a new canonical innovation representation of X, using linear filtering theory. When the kernel K(s,t) = partial derivative(2)/partial derivative s partial derivative t EG(t)G(s,) s not equal t has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with additional fractional structure, including the mixed fractional Brownian motion from mathematical finance. We show how previously-known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way, and complement them with new formulas for Radon-Nikodym densities.