Weierstrass bridges

Schied, Alexander; Zhang, Zhenyuan

doi:10.1090/tran/9116

Weierstrass bridges
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by Alexander Schied and Zhenyuan Zhang

Trans. Amer. Math. Soc. 377 (2024), 2947-2989

DOI: https://doi.org/10.1090/tran/9116

Published electronically: February 20, 2024

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Abstract:

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 a 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.

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