Exact simulation of the first-passage time of SDEs to time-dependent thresholds
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915072517865472 |
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| author | Khurana, Devika Desmettre, Sascha Buckwar, Evelyn |
| author_facet | Khurana, Devika Desmettre, Sascha Buckwar, Evelyn |
| contents | The first-passage time (FPT) is a fundamental concept in stochastic processes, representing the time it takes for a process to reach a specified threshold for the first time. Often, considering a time-dependent threshold is essential for accurately modeling stochastic processes, as it provides a more accurate and adaptable framework. In this paper, we extend an existing Exact simulation method developed for constant thresholds to handle time-dependent thresholds. Our proposed approach utilizes the FPT of Brownian motion and accepts it for the FPT of a given process with some probability, which is determined using Girsanov's transformation. This method eliminates the need to simulate entire paths over specific time intervals, avoids time-discretization errors, and directly simulates the first-passage time. We present results demonstrating the method's effectiveness, including the extension to time-dependent thresholds, an analysis of its time complexity, comparisons with existing methods through numerical examples, and its application to predicting spike times in a neuron. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact simulation of the first-passage time of SDEs to time-dependent thresholds Khurana, Devika Desmettre, Sascha Buckwar, Evelyn Probability Numerical Analysis 37M05, 65C30, 60G05, 60H35, 68Q87, 92C20 The first-passage time (FPT) is a fundamental concept in stochastic processes, representing the time it takes for a process to reach a specified threshold for the first time. Often, considering a time-dependent threshold is essential for accurately modeling stochastic processes, as it provides a more accurate and adaptable framework. In this paper, we extend an existing Exact simulation method developed for constant thresholds to handle time-dependent thresholds. Our proposed approach utilizes the FPT of Brownian motion and accepts it for the FPT of a given process with some probability, which is determined using Girsanov's transformation. This method eliminates the need to simulate entire paths over specific time intervals, avoids time-discretization errors, and directly simulates the first-passage time. We present results demonstrating the method's effectiveness, including the extension to time-dependent thresholds, an analysis of its time complexity, comparisons with existing methods through numerical examples, and its application to predicting spike times in a neuron. |
| title | Exact simulation of the first-passage time of SDEs to time-dependent thresholds |
| topic | Probability Numerical Analysis 37M05, 65C30, 60G05, 60H35, 68Q87, 92C20 |
| url | https://arxiv.org/abs/2412.13060 |