Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866908634250739712 |
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| author | Qian, Zhongmin Xu, Xingcheng |
| author_facet | Qian, Zhongmin Xu, Xingcheng |
| contents | This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters \( H \in (\frac{1}{3}, \frac{1}{2}] \), using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1803_00335 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion Qian, Zhongmin Xu, Xingcheng Probability 60H05, 60H10, 91B70, 91G80 This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters \( H \in (\frac{1}{3}, \frac{1}{2}] \), using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling. |
| title | Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion |
| topic | Probability 60H05, 60H10, 91B70, 91G80 |
| url | https://arxiv.org/abs/1803.00335 |