Occupation time, quantile and rank of the Ornstein–Uhlenbeck process and their applications to mathematical finance

Yuri Imamura; Takashi Kato; Ryozo Miura; Ju-Yi Yen

doi:10.3934/fmf.2025007

Article Contents

June 2025, Volume 5, 121-139. Doi: 10.3934/fmf.2025007 open access

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Occupation time, quantile and rank of the Ornstein–Uhlenbeck process and their applications to mathematical finance

1.
Department of Business Economics, School of Management, Tokyo University of Science, 1–11–2, Fujimi, Chiyoda-ku, Tokyo 102-0071, Japan
2.
Association of Mathematical Finance Laboratory (AMFiL), 2–15, Minamiaoyama, Minato-ku, Tokyo 107-0062, Japan
3.
School of Business Administration, Hitotsubashi University Business School, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8439, Japan
4.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA

^*Corresponding author: Ju-Yi Yen
^*Corresponding author: Ju-Yi Yen

Received: December 2024

Revised: May 2025

Published online: June 12, 2025

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Abstract

This study examines critical stochastic characteristics of the Ornstein–Uhlenbeck (OU) process, including occupation time, quantile, and rank, with applications in mathematical finance. These metrics provide essential insights into the behavior of stochastic systems, enabling more refined financial modeling and risk assessment. We develop a mathematical framework for analyzing the occupation time and quantile of the OU process using Girsanov transformations and Laplace transform techniques, extending prior results rooted in geometric Brownian motion. Additionally, we investigate the rank process, which defines stochastic levels, offering a perspective on its potential financial applications. The theoretical results are complemented by asymptotic expansions and numerical validations, demonstrating their relevance to derivative pricing, risk management, and other financial contexts. This work bridges foundational stochastic process theory with practical tools for addressing complex challenges in modern finance.

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Mathematics Subject Classification: Primary: 60J60, 91G80, 91G20; Secondary: 60G44.

Citation:

FullText(HTML)

Table 1. Pattern 1 ($ \theta = 0.5, \ \ a = 0.5, \ \ \sigma = 0.3, \ \ K = 0.6, \ \ T = 1, \ \ x = 0.4 $)

$ \lambda $	Benchmark	$ v_\lambda(x) $	RE
0.2	0.20212	0.20210	$ -1.3\times 10^{-4} $
0.4	0.18798	0.18784	$ -7.5\times 10^{-4} $
0.6	0.17402	0.17358	$ -2.5\times 10^{-3} $
0.8	0.16035	0.15933	$ -6.4\times 10^{-3} $
1.0	0.14702	0.14507	$ -1.3\times 10^{-2} $

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Table 2. Pattern 2 ($ \theta = 0.7, \ \ a = 0.4, \ \ \sigma = 0.5, \ \ K = 0.6, \ \ T = 1, \ \ x = 0.5 $)

$ \lambda $	Benchmark	$ v_\lambda(x) $	RE
0.2	0.25377	0.25351	$ -1.0\times 10^{-3} $
0.4	0.27063	0.26954	$ -4.0\times 10^{-3} $
0.6	0.28795	0.28558	$ -8.2\times 10^{-3} $
0.8	0.30567	0.30162	$ -1.3\times 10^{-2} $
1.0	0.32369	0.31765	$ -1.9\times 10^{-2} $

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