Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, α}$-domains has finite radius of convergence everywhere: $2\leq d \leq 8$
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910028251791360 |
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| author | Li, Siran Ni, Hao |
| author_facet | Li, Siran Ni, Hao |
| contents | A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $Ω$, it is only known that the radius of convergence for the expected signature on sufficiently regular $Ω$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $Ω$ is the $2$-dimensional unit disc ([1]).
In this paper, we prove that on any bounded $C^{2,α}$-domain $Ω\subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_07917 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, α}$-domains has finite radius of convergence everywhere: $2\leq d \leq 8$ Li, Siran Ni, Hao Probability Analysis of PDEs Classical Analysis and ODEs 60L20, 35R45 A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $Ω$, it is only known that the radius of convergence for the expected signature on sufficiently regular $Ω$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $Ω$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,α}$-domain $Ω\subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains. |
| title | Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, α}$-domains has finite radius of convergence everywhere: $2\leq d \leq 8$ |
| topic | Probability Analysis of PDEs Classical Analysis and ODEs 60L20, 35R45 |
| url | https://arxiv.org/abs/2011.07917 |