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Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent

Frank Aurzada, Sabine Müller

Abstract

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.

Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent

Abstract

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 , it is well known that the persistence probability of an FBM below a constant barrier decays like , as tends to infinity, cf. Molchan (1999). Our object of interest is the persistence probability of the process resulting from first randomly selecting and then considering a fractional Brownian motion with this value of 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 , as tends to infinity, where is the essential supremum of the distribution of the random Hurst exponent.
Paper Structure (14 sections, 17 theorems, 139 equations, 1 figure)

This paper contains 14 sections, 17 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction and Main Result
  2. Monotonicity in $H$.
  3. Multi-parameter FBM with random Hurst exponent.
  4. Other Generalizations.
  5. Outline.
  6. Proof of Theorem \ref{['thm:smallbarrier']}
  7. Proof of the Lower Bound in Theorem \ref{['Thm']}
  8. Preliminaries
  9. A Uniform Version of Statement 1 in Molchan (1999)
  10. Lower Bound
  11. Proof of the Upper Bound in Theorem \ref{['Thm']}
  12. Lemmas for the Upper Bound
  13. A uniform upper bound for the persistence probability
  14. Upper Bound

Key Result

Theorem 1.1

Let $(B_t^\mathcal{H})_{t\geq 0}$ be an FBMRE. Then

Figures (1)

  • Figure 1: Sketch of the right-to-left record times $R_1^H<\cdots<R_{T_n^H+1}^H$, with reference points at $n^2$ and $n^2+n$ for $H = 1/2$, $n = 15$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['thm:smallbarrier']}
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 18 more