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Distributional and Extremal Behaviour of Brownian Motion with Exponential Resetting

Krzysztof Dębicki, Enkelejd Hashorva, Zbigniew Michna

Abstract

We study the distributional and asymptotic properties of the supremum of Brownian motion with drift and exponential resetting. We obtain an explicit renewal-type formula for the distribution of the supremum and then derive an approximation for its survival function. Moreover, we find the asymptotics of the tail distribution of the infimum. We also consider the stationary case and give a new explicit expression for the fidi's of such processes.

Distributional and Extremal Behaviour of Brownian Motion with Exponential Resetting

Abstract

We study the distributional and asymptotic properties of the supremum of Brownian motion with drift and exponential resetting. We obtain an explicit renewal-type formula for the distribution of the supremum and then derive an approximation for its survival function. Moreover, we find the asymptotics of the tail distribution of the infimum. We also consider the stationary case and give a new explicit expression for the fidi's of such processes.
Paper Structure (13 sections, 9 theorems, 139 equations, 2 figures, 3 tables)

This paper contains 13 sections, 9 theorems, 139 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Distributional properties of Brownian motion with resetting
  3. Extremal behaviour of Brownian motion with resetting
  4. Stationary Brownian motion with resetting
  5. Numerical examples
  6. Appendix
  7. Proof of Theorem \ref{['jointdist_drift']}
  8. Proof of Theorem \ref{['th2']}
  9. Proof of Theorem \ref{['th.sup.as']}
  10. Proof of Theorem \ref{['thm:BM-window-reset-general-x0']}
  11. Proof of Proposition \ref{['prop.stat']}
  12. Proof of Theorem \ref{['thasimpBMS']}
  13. Proof of Theorem \ref{['thasympSBMl']}

Key Result

Theorem 1

Assume that $X^{(c)}_0=x_0$, where $x_0$ is a real-valued random variable independent of $(W,N)$. For $0\le s<t$ and $u,w\in\mathbb{R}$,

Figures (2)

  • Figure 1: A trajectory of Brownian motion with resetting with $c=1$, $x_0=0$, $x_R=1$, $\lambda=2$. $U^{(1)},\, U^{(2)},\,U^{(3)}$ are epochs of resetting.
  • Figure 2: Cumulative distribution function of $\sup_{t\in [0,1]} X_t$ for $x_R=0$, $\sigma=1$ and $\lambda=0.1, 0.5, 1, 2, 3, 5$.

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Proposition 2
  • ...and 3 more