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arXiv · 2403.05409

Simulating conditioned diffusions on manifolds

Marc Corstanje·Frank van der Meulen·Moritz Schauer·Stefan Sommer
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Abstract

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.

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Marc Corstanje, Frank van der Meulen, Moritz Schauer, Stefan Sommer. 2025-02-21. Simulating conditioned diffusions on manifolds. https://arxiv.org/abs/2403.05409

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