On first passage time problems of Brownian motion -- The inverse method of images revisited
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911853270007808 |
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| author | Christensen, Sören Hallmann, Oskar Klein, Maike |
| author_facet | Christensen, Sören Hallmann, Oskar Klein, Maike |
| contents | Let $W$ be a standard Brownian motion with $W_0 = 0$ and let $b\colon[0,\infty) \to \mathbb{R}$ be a continuous function with $b(0) > 0$. In this article, we look at the classical First Passage Time (FPT) problem, i.e., the question of determining the distribution of $τ:= \inf \{ t\in [0,\infty)\colon W_t \geq b(t) \}.$ More specifically, we revisit the method of images, which we feel has received less attention than it deserves. The main observation of this approach is that the FPT problem is fully solved if a measure $μ$ exists such that \begin{align*} \int_{(0,\infty)} \exp\left(-\frac{θ^2}{2t}+\frac{θb(t)}{t}\right)μ(dθ)=1, \qquad t\in(0,\infty). \end{align*} The goal of this article is to lay the foundation for answering the still open question of the existence and characterisation of such a measure $μ$ for a given curve $b$. We present a new duality approach that allows us to give sufficient conditions for the existence. Moreover, we introduce a very efficient algorithm for approximating the representing measure $μ$ and provide a rigorous theoretical foundation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16615 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On first passage time problems of Brownian motion -- The inverse method of images revisited Christensen, Sören Hallmann, Oskar Klein, Maike Probability 90C05, 60G40, 60J65 Let $W$ be a standard Brownian motion with $W_0 = 0$ and let $b\colon[0,\infty) \to \mathbb{R}$ be a continuous function with $b(0) > 0$. In this article, we look at the classical First Passage Time (FPT) problem, i.e., the question of determining the distribution of $τ:= \inf \{ t\in [0,\infty)\colon W_t \geq b(t) \}.$ More specifically, we revisit the method of images, which we feel has received less attention than it deserves. The main observation of this approach is that the FPT problem is fully solved if a measure $μ$ exists such that \begin{align*} \int_{(0,\infty)} \exp\left(-\frac{θ^2}{2t}+\frac{θb(t)}{t}\right)μ(dθ)=1, \qquad t\in(0,\infty). \end{align*} The goal of this article is to lay the foundation for answering the still open question of the existence and characterisation of such a measure $μ$ for a given curve $b$. We present a new duality approach that allows us to give sufficient conditions for the existence. Moreover, we introduce a very efficient algorithm for approximating the representing measure $μ$ and provide a rigorous theoretical foundation. |
| title | On first passage time problems of Brownian motion -- The inverse method of images revisited |
| topic | Probability 90C05, 60G40, 60J65 |
| url | https://arxiv.org/abs/2404.16615 |