Simulating conditioned diffusions on manifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916623363866624 |
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| author | Corstanje, Marc van der Meulen, Frank Schauer, Moritz Sommer, Stefan |
| author_facet | Corstanje, Marc van der Meulen, Frank Schauer, Moritz Sommer, Stefan |
| contents | To date, most methods for simulating conditioned diffusions are limited to the Euclidean setting. The conditioned process can be constructed using a change of measure known as Doob's $h$-transform. The specific type of conditioning depends on a function $h$ which is typically unknown in closed form. To resolve this, we extend the notion of guided processes to a manifold $M$, where one replaces $h$ by a function based on the heat kernel on $M$. We consider the case of a Brownian motion with drift, constructed using the frame bundle of $M$, conditioned to hit a point $x_T$ at time $T$. We prove equivalence of the laws of the conditioned process and the guided process with a tractable Radon-Nikodym derivative. Subsequently, we show how one can obtain guided processes on any manifold $N$ that is diffeomorphic to $M$ without assuming knowledge of the heat kernel on $N$. We illustrate our results with numerical simulations of guided processes and Bayesian parameter estimation based on discrete-time observations. For this, we consider both the torus and the Poincaré disk. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05409 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simulating conditioned diffusions on manifolds Corstanje, Marc van der Meulen, Frank Schauer, Moritz Sommer, Stefan Probability Statistics Theory 62R30, 60J60, 60J25 To date, most methods for simulating conditioned diffusions are limited to the Euclidean setting. The conditioned process can be constructed using a change of measure known as Doob's $h$-transform. The specific type of conditioning depends on a function $h$ which is typically unknown in closed form. To resolve this, we extend the notion of guided processes to a manifold $M$, where one replaces $h$ by a function based on the heat kernel on $M$. We consider the case of a Brownian motion with drift, constructed using the frame bundle of $M$, conditioned to hit a point $x_T$ at time $T$. We prove equivalence of the laws of the conditioned process and the guided process with a tractable Radon-Nikodym derivative. Subsequently, we show how one can obtain guided processes on any manifold $N$ that is diffeomorphic to $M$ without assuming knowledge of the heat kernel on $N$. We illustrate our results with numerical simulations of guided processes and Bayesian parameter estimation based on discrete-time observations. For this, we consider both the torus and the Poincaré disk. |
| title | Simulating conditioned diffusions on manifolds |
| topic | Probability Statistics Theory 62R30, 60J60, 60J25 |
| url | https://arxiv.org/abs/2403.05409 |