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Online minimum search for Brownian motion and the Cauchy process: Multiple approaches

Shannon Starr, Erik Wu

Abstract

The distribution for the minimum of Brownian motion or the Cauchy process is well-known using the reflection principle. Here we consider the problem of finding the sample-by-sample minimum, which we call the online minimum search. We consider the possibility of the golden search method, but we show quantitatively that the bisection method is more efficient. In the bisection method there is a hierarchical parameter, which tunes the depth to which each sub-search is conducted, somewhat similarly to how a depth-first search works to generate a topological ordering on nodes. Finally, we consider the possibility of using harmonic measure, which is a novel idea that has so far been unexplored.

Online minimum search for Brownian motion and the Cauchy process: Multiple approaches

Abstract

The distribution for the minimum of Brownian motion or the Cauchy process is well-known using the reflection principle. Here we consider the problem of finding the sample-by-sample minimum, which we call the online minimum search. We consider the possibility of the golden search method, but we show quantitatively that the bisection method is more efficient. In the bisection method there is a hierarchical parameter, which tunes the depth to which each sub-search is conducted, somewhat similarly to how a depth-first search works to generate a topological ordering on nodes. Finally, we consider the possibility of using harmonic measure, which is a novel idea that has so far been unexplored.
Paper Structure (21 sections, 2 theorems, 48 equations, 18 figures, 3 tables)

This paper contains 21 sections, 2 theorems, 48 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Potential Search Methods
  3. Golden Section Search
  4. Iterative Golden Section Search
  5. Analyzing the Iterative Golden Section Search and the Naive Approach
  6. Monte Carlo Bisection Method
  7. Monte Carlo Bisection on a Brownian Bridge and Cauchy Process Path
  8. Comparison Between Search Methods
  9. Accuracy for Iterative GSS vs. MCB
  10. Efficiency for Iterative GSS vs. MCB
  11. Conformal mappings and the Harmonic measure
  12. Hitting distribution on a bridge for reflected B.m. on a strip
  13. The Schwarz-Christoffel formula and SC toolbox
  14. Applications and Outlook
  15. A hybrid online search model
  16. ...and 6 more sections

Key Result

Theorem 4.1

Consider the unique conformal mapping $\varphi : \mathbb{H}^+ \to \mathcal{U}$ which is a bijection and such that for the continuous extension to the boundaries, the homeomorphism This exists by Caratheodory's theorem. See for example Krantz. satisfies: $\infty$ is mapped to $\infty$ and $[0,1]$ is

Figures (18)

  • Figure 1: A symmetric number line representing the first iteration of a GSS
  • Figure 2: A possible graph when $f(t_2) > f(t_1)$
  • Figure 3: A possible graph when $f(t_1) > f(t_2)$
  • Figure 4: A sample GSS algorithm call on $f(x) = \cos(x)$
  • Figure 5: A sample GSS algorithm call on $f(x) = x + \sin(x) + x\sin(x)$
  • ...and 13 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • Corollary 4.2