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Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary

Yoann Potiron

Abstract

An explicit formula for the probability that a continuous local martingale crosses a one or two-sided random constant boundary in a finite time interval is derived. We obtain that the boundary crossing probability of a continuous local martingale to a constant boundary is equal to the boundary crossing probability of a standard Wiener process to a constant boundary up to a time change of quadratic variation value. This relies on the constancy of the boundary and the Dambis, Dubins-Schwarz theorem for continuous local martingale. The main idea of the proof is the scale invariant property of the time-changed Wiener process and thus the scale invariant property of the first-passage time. As an application, we also consider an inverse first-passage time problem of quadratic variation.

Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary

Abstract

An explicit formula for the probability that a continuous local martingale crosses a one or two-sided random constant boundary in a finite time interval is derived. We obtain that the boundary crossing probability of a continuous local martingale to a constant boundary is equal to the boundary crossing probability of a standard Wiener process to a constant boundary up to a time change of quadratic variation value. This relies on the constancy of the boundary and the Dambis, Dubins-Schwarz theorem for continuous local martingale. The main idea of the proof is the scale invariant property of the time-changed Wiener process and thus the scale invariant property of the first-passage time. As an application, we also consider an inverse first-passage time problem of quadratic variation.
Paper Structure (26 sections, 42 theorems, 214 equations)

This paper contains 26 sections, 42 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Problem
  3. Motivation
  4. Existing results
  5. This paper
  6. Structure of this paper
  7. Explicit formula
  8. One-sided (i) case
  9. Two-sided (i) case
  10. One-sided (ii) case
  11. Two-sided (ii) case
  12. Applications in the inverse FPT problem
  13. One-sided (i) case
  14. Two-sided (i) case
  15. One-sided (ii) case
  16. ...and 11 more sections

Key Result

Lemma 2.1

We obtain a Levy distribution with $P_g^W (0) = 0$, $f_g^W (0) = 0$,

Theorems & Definitions (114)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Example 1
  • Proposition 2.1
  • Theorem 2.3
  • Corollary 2.1
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.2
  • ...and 104 more