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On the Absolute-Value Integral of a Brownian Motion with Drift: Exact and Asymptotic Formulae

Weixuan Xia, Yuyang Zhang

TL;DR

This work analyzes the distribution of the drifted Brownian integral $X_t=\int_0^t|\mu s+\sigma W_s|\,ds$ and develops a comprehensive set of exact and asymptotic results. An asymptotic expansion of the space Laplace transform yields rapidly convergent series for the density and distribution in terms of Meijer’s $G$-function and hypergeometric functions; a recursive scheme provides exact computation of moments, which collapse to exponentials and Gauss’ error function after simplification. A marginal space-time transform together with a generalized Laplace method for exponential Airy integrals delivers sharp small- and large-deviation estimates and illuminates the impact of drift on the full distribution. The results extend the theory of Brownian functionals with drift, enabling accurate, implementable computations for applications in reliability and financial modeling, among others.

Abstract

The present paper is concerned with the integral of the absolute value of a Brownian motion with drift. By establishing an asymptotic expansion of the space Laplace transform, we obtain series representations for the probability density function and cumulative distribution function of the integral, making use of Meijer's G-function. A functional recursive formula is derived for the moments, which is shown to yield only exponentials and Gauss' error function up to arbitrary orders, permitting exact computations. To obtain sharp asymptotic estimates for small- and large-deviation probabilities, we employ a marginal space-time Laplace transform and apply a newly developed generalization of Laplace's method to exponential Airy integrals. The impact of drift on the complete distribution of the integral is explored in depth. The resultant new formulae complement existing ones in the standard Brownian motion case to great extent in terms of both theoretical generality and modeling capacity and have been presented for easy implementation, which numerical experiments demonstrate.

On the Absolute-Value Integral of a Brownian Motion with Drift: Exact and Asymptotic Formulae

TL;DR

This work analyzes the distribution of the drifted Brownian integral and develops a comprehensive set of exact and asymptotic results. An asymptotic expansion of the space Laplace transform yields rapidly convergent series for the density and distribution in terms of Meijer’s -function and hypergeometric functions; a recursive scheme provides exact computation of moments, which collapse to exponentials and Gauss’ error function after simplification. A marginal space-time transform together with a generalized Laplace method for exponential Airy integrals delivers sharp small- and large-deviation estimates and illuminates the impact of drift on the full distribution. The results extend the theory of Brownian functionals with drift, enabling accurate, implementable computations for applications in reliability and financial modeling, among others.

Abstract

The present paper is concerned with the integral of the absolute value of a Brownian motion with drift. By establishing an asymptotic expansion of the space Laplace transform, we obtain series representations for the probability density function and cumulative distribution function of the integral, making use of Meijer's G-function. A functional recursive formula is derived for the moments, which is shown to yield only exponentials and Gauss' error function up to arbitrary orders, permitting exact computations. To obtain sharp asymptotic estimates for small- and large-deviation probabilities, we employ a marginal space-time Laplace transform and apply a newly developed generalization of Laplace's method to exponential Airy integrals. The impact of drift on the complete distribution of the integral is explored in depth. The resultant new formulae complement existing ones in the standard Brownian motion case to great extent in terms of both theoretical generality and modeling capacity and have been presented for easy implementation, which numerical experiments demonstrate.
Paper Structure (7 sections, 14 theorems, 133 equations, 5 figures, 4 tables)

This paper contains 7 sections, 14 theorems, 133 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Laplace transform
  3. Distribution functions
  4. Moments
  5. Asymptotic estimates
  6. Numerical experiments
  7. Concluding remarks

Key Result

Theorem 1

Assume $\mu\neq0$. The space Laplace transform (2.2) of $X_{t}$ admits the following series representation: where $\mathrm{Ai}\equiv\mathrm{Ai}(\cdot)$ is the first Airy function and $\{\alpha'_{k}:k\in\mathds{N}_{++}\}\subset\mathds{R}_{--}$ are the zeros of its derivative, ordered in such a way that $\alpha'_{k}>\alpha'_{k+1}$, $\forall k$, $\mathrm{\Gamma}\equiv\mathrm{\Gamma}(\cdot)$ is the u

Figures (5)

  • Figure 1: Drift impact on skewness and excess kurtosis
  • Figure 2: Probability density function and cumulative distribution function (I)
  • Figure 3: Probability density function and cumulative distribution function (II)
  • Figure 4: Probability density function and cumulative distribution function (III)
  • Figure 5: Probability density function and cumulative distribution function (IV)

Theorems & Definitions (28)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Theorem 4
  • proof
  • ...and 18 more