A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913813082669056 |
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| author | Becher, Mrabet Boudabra, Maher Haggui, Fathi |
| author_facet | Becher, Mrabet Boudabra, Maher Haggui, Fathi |
| contents | We present a numerical framework to approximate the $μ$-domain in the planar Skorokhod embedding problem (PSEP), recently appeared in \cite{gross2019}. Our approach investigates the continuity and convergence properties of the solutions with respect to the underlying distribution $μ$. We establish that, under weak convergence of a sequence of probability measures $(μ_n)$ with bounded support, the corresponding sequence of $μ_n$-domains converges to the domain associated with $μ$, limit of $(μ_n)$. We derive explicit convergence results in the $L^1$ norm, supported by a generalization using the concept of $α_p$-convergence. Furthermore, we provide practical implementation techniques, convergence rate estimates, and numerical simulations using various distributions. The method proves robust and adaptable, offering a concrete computational pathway for approximating $μ$-domains in the PSEP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem Becher, Mrabet Boudabra, Maher Haggui, Fathi Probability We present a numerical framework to approximate the $μ$-domain in the planar Skorokhod embedding problem (PSEP), recently appeared in \cite{gross2019}. Our approach investigates the continuity and convergence properties of the solutions with respect to the underlying distribution $μ$. We establish that, under weak convergence of a sequence of probability measures $(μ_n)$ with bounded support, the corresponding sequence of $μ_n$-domains converges to the domain associated with $μ$, limit of $(μ_n)$. We derive explicit convergence results in the $L^1$ norm, supported by a generalization using the concept of $α_p$-convergence. Furthermore, we provide practical implementation techniques, convergence rate estimates, and numerical simulations using various distributions. The method proves robust and adaptable, offering a concrete computational pathway for approximating $μ$-domains in the PSEP. |
| title | A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem |
| topic | Probability |
| url | https://arxiv.org/abs/2504.21531 |