Weierstrass Bridges
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917556765327360 |
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| author | Schied, Alexander Zhang, Zhenyuan |
| author_facet | Schied, Alexander Zhang, Zhenyuan |
| contents | We introduce a new class of stochastic processes called fractional Wiener-Weierstrass bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstrass functions to an underlying fractional Brownian bridge. By analyzing the $p$-th variation of the fractional Wiener-Weierstrass bridge along the sequence of $b$-adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener-Weierstrass bridges that are based on standard Brownian bridge. We furthermore prove that fractional Wiener-Weierstrass bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstrass bridges based on bridges derived from a general continuous Gaussian martingale. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04944 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weierstrass Bridges Schied, Alexander Zhang, Zhenyuan Probability 60G22, 60G15, 60G17, 28A80 We introduce a new class of stochastic processes called fractional Wiener-Weierstrass bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstrass functions to an underlying fractional Brownian bridge. By analyzing the $p$-th variation of the fractional Wiener-Weierstrass bridge along the sequence of $b$-adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener-Weierstrass bridges that are based on standard Brownian bridge. We furthermore prove that fractional Wiener-Weierstrass bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstrass bridges based on bridges derived from a general continuous Gaussian martingale. |
| title | Weierstrass Bridges |
| topic | Probability 60G22, 60G15, 60G17, 28A80 |
| url | https://arxiv.org/abs/2304.04944 |