Convex hull of Brownian motion and Brownian bridge
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917692132294656 |
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| author | Šebek, Stjepan |
| author_facet | Šebek, Stjepan |
| contents | In this article we study the convex hull spanned by the union of trajectories of a standard planar Brownian motion, and an independent standard planar Brownian bridge. We find exact values of the expectation of perimeter and area of such a convex hull. As an auxiliary result, that is of interest in its own right, we provide an explicit shape of the probability density function of a random variable that represents the time when combined maximum of a standard one-dimensional Brownian motion, and an independent standard one-dimensional Brownian bridge is attained. At the end, we generalize our results to the case of multiple independent standard planar Brownian motions and Brownian bridges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convex hull of Brownian motion and Brownian bridge Šebek, Stjepan Probability 60J65, 60D05, 52A22, 52A10 In this article we study the convex hull spanned by the union of trajectories of a standard planar Brownian motion, and an independent standard planar Brownian bridge. We find exact values of the expectation of perimeter and area of such a convex hull. As an auxiliary result, that is of interest in its own right, we provide an explicit shape of the probability density function of a random variable that represents the time when combined maximum of a standard one-dimensional Brownian motion, and an independent standard one-dimensional Brownian bridge is attained. At the end, we generalize our results to the case of multiple independent standard planar Brownian motions and Brownian bridges. |
| title | Convex hull of Brownian motion and Brownian bridge |
| topic | Probability 60J65, 60D05, 52A22, 52A10 |
| url | https://arxiv.org/abs/2406.07079 |