Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908888206409728 |
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| author | Aurzada, Frank Müller, Sabine |
| author_facet | Aurzada, Frank Müller, Sabine |
| contents | We study the persistence properties of a fractional Brownian motion whose Hurst exponent is a random variable instead of a fixed constant. For each fixed $H \in (0,1)$, it is well known that the persistence probability of an FBM below a constant barrier decays like $T^{-(1-H)+o(1)}$, as $T$ tends to infinity, cf. Molchan (1999). Our object of interest is the persistence probability of the process resulting from first randomly selecting $H\in (0,1)$ and then considering a fractional Brownian motion with this value of $H$ as a Hurst exponent, a process that is referred to as a fractional Brownian motion with random exponent. We prove that its persistence probability decays as $T^{-(1-H_0)+o(1)}$, as $T$ tends to infinity, where $H_0$ is the essential supremum of the distribution of the random Hurst exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14934 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent Aurzada, Frank Müller, Sabine Probability We study the persistence properties of a fractional Brownian motion whose Hurst exponent is a random variable instead of a fixed constant. For each fixed $H \in (0,1)$, it is well known that the persistence probability of an FBM below a constant barrier decays like $T^{-(1-H)+o(1)}$, as $T$ tends to infinity, cf. Molchan (1999). Our object of interest is the persistence probability of the process resulting from first randomly selecting $H\in (0,1)$ and then considering a fractional Brownian motion with this value of $H$ as a Hurst exponent, a process that is referred to as a fractional Brownian motion with random exponent. We prove that its persistence probability decays as $T^{-(1-H_0)+o(1)}$, as $T$ tends to infinity, where $H_0$ is the essential supremum of the distribution of the random Hurst exponent. |
| title | Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent |
| topic | Probability |
| url | https://arxiv.org/abs/2603.14934 |