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Explicit Asymptotics on First Passage Times of Diffusion Processes

Angelos Dassios, Luting Li

TL;DR

This work develops a unified perturbation framework for explicit first passage time densities of time-homogeneous diffusions by formulating a killed Dirichlet problem in the Laplace domain and solving a recursive sequence in the frequency domain. The method yields closed-form asymptotic FPT densities for single-sided level crossings, with an exact probabilistic representation of truncation error and rigorous convergence properties that hold beyond small perturbations. Application to Ornstein–Uhlenbeck and Bessel processes provides concrete Nth-order densities expressed via parabolic-cylinder functions, along with detailed left/right tail behavior and error terms. Numerical experiments with downward OU FPTD demonstrate strong accuracy and substantial computational advantages over conventional inverse-Laplace techniques, while highlighting limitations in long-time tail regions for larger orders. The framework further extends to time-changed diffusions (e.g., exponential-Shiryaev/CIR-type models) and offers practical tools for survival analysis and finance where closed-form FPTs are valuable.

Abstract

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. According to the killed version potential theory and the perturbation theory, we are able to deduce closed-form solutions for probability densities of single-sided level crossing problem. The framework is applicable to diffusion processes with continuous drift functions, and a recursive system in the frequency domain has been provided. Besides, we derive a probabilistic representation for error estimation. The representation can be used to evaluate deviations in perturbed density functions. In the present paper, we apply the framework to Ornstein-Uhlenbeck and Bessel processes to find closed-form approximations for their first passage times; another successful application is given by the exponential-Shiryaev process. Numerical results are provided at the end of this paper.

Explicit Asymptotics on First Passage Times of Diffusion Processes

TL;DR

This work develops a unified perturbation framework for explicit first passage time densities of time-homogeneous diffusions by formulating a killed Dirichlet problem in the Laplace domain and solving a recursive sequence in the frequency domain. The method yields closed-form asymptotic FPT densities for single-sided level crossings, with an exact probabilistic representation of truncation error and rigorous convergence properties that hold beyond small perturbations. Application to Ornstein–Uhlenbeck and Bessel processes provides concrete Nth-order densities expressed via parabolic-cylinder functions, along with detailed left/right tail behavior and error terms. Numerical experiments with downward OU FPTD demonstrate strong accuracy and substantial computational advantages over conventional inverse-Laplace techniques, while highlighting limitations in long-time tail regions for larger orders. The framework further extends to time-changed diffusions (e.g., exponential-Shiryaev/CIR-type models) and offers practical tools for survival analysis and finance where closed-form FPTs are valuable.

Abstract

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. According to the killed version potential theory and the perturbation theory, we are able to deduce closed-form solutions for probability densities of single-sided level crossing problem. The framework is applicable to diffusion processes with continuous drift functions, and a recursive system in the frequency domain has been provided. Besides, we derive a probabilistic representation for error estimation. The representation can be used to evaluate deviations in perturbed density functions. In the present paper, we apply the framework to Ornstein-Uhlenbeck and Bessel processes to find closed-form approximations for their first passage times; another successful application is given by the exponential-Shiryaev process. Numerical results are provided at the end of this paper.

Paper Structure

This paper contains 32 sections, 17 theorems, 159 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Perturbed Dirichlet Problem
  4. Truncation Error and Convergence
  5. Proof:
  6. Recursions under Frequency Domain
  7. Proof:
  8. Proof:
  9. Ornstein-Uhlenbeck Process
  10. Proof:
  11. Proof:
  12. Proof:
  13. Proof:
  14. Proof:
  15. Proof:
  16. ...and 17 more sections

Key Result

Proposition 2.1

For all $t\in (0,+\infty)$ and all $\beta\in\mathbb{C}$ with $Real(\beta)>0$, if then Further, if for some constant $M<+\infty$ then

Figures (16)

  • Figure 1: FPT density of OU process
  • Figure 2: Relative error to iv)
  • Figure 3: FPTD of OU process for general case
  • Figure 4: Relative error to i) for general case
  • Figure 5: OU left tail density for $l= \theta$
  • ...and 11 more figures

Theorems & Definitions (22)

  • Proposition 2.1: Probabilistic Representation for the Truncation Error
  • Remark 2.2
  • Lemma 2.3: Laplace Transform for the FPTD of Brownian Motion
  • Lemma 2.4: Recursive Solution to $o\left(\epsilon^i\right)$
  • Remark 2.5
  • Proposition 3.1: Recursive Polynomial
  • Lemma 3.2: Recursive Polynomial Decomposition
  • Remark 3.3
  • Proposition 3.4: $N$-th Order Perturbed FPTD of OU Process
  • Remark 3.5
  • ...and 12 more