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Weierstrass Bridges

Alexander Schied, Zhenyuan Zhang

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 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.

Weierstrass Bridges

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 -th variation of the fractional Wiener-Weierstrass bridge along the sequence of -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.
Paper Structure (13 sections, 29 theorems, 199 equations, 2 figures)

This paper contains 13 sections, 29 theorems, 199 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Outlook and some open questions.
  4. Proofs
  5. Proof of \ref{['main thm']} (a)
  6. Proof of \ref{['main thm']} (b)
  7. Convergence of the expectation
  8. A concentration bound
  9. Linearity of the $p$-th variation
  10. Proof of \ref{['critical case H=1/2 thm']}
  11. Proof of \ref{['semimartingale thm']}
  12. Other proofs
  13. Fractal functions with Hölder continuous base

Key Result

Theorem 2.3

Let $X$ be a fractional Wiener--Weierstraß bridge with parameters $\alpha$, $b$, and $H$, and suppose that $H\neq K=1\wedge(-\log_b\alpha)$. Then ${\mathbb P}$-almost every sample path of $X$ admits the roughness exponent $H\wedge K$. More precisely:

Figures (2)

  • Figure 1: Illustration of \ref{['main thm']} (a) by means of histograms of the $(1/K)^{\text{th}}$ variation $\sum_{k=0}^{b^n}|X((k+1)b^{-n})-X(kb^{-n})|^{1/K}$ for 5000 sample paths of the fractional Wiener--Weierstraß bridge with $n=16$, $b=2$, and parameters $H=0.7$ and $K=0.5$ (left) versus $H=0.5$ and $K=0.2$ (right).
  • Figure 2: Covariance functions of the Wiener--Weierstraß bridge for $H=1/2$, $\alpha=1/2$, and $b=2$ (left) and $b=3$ (right).

Theorems & Definitions (64)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Remark 2.8
  • Proposition 2.9
  • Lemma 3.1
  • ...and 54 more