New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910757408473088 |
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| author | Takahashi, Akihiko Yamada, Toshihiro |
| author_facet | Takahashi, Akihiko Yamada, Toshihiro |
| contents | This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index $H<1/2$, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13405 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion Takahashi, Akihiko Yamada, Toshihiro Probability Numerical Analysis 60G22, 60H07, 60L20 This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index $H<1/2$, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. |
| title | New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion |
| topic | Probability Numerical Analysis 60G22, 60H07, 60L20 |
| url | https://arxiv.org/abs/2306.13405 |