Abstract
We present a simulation scheme for simulating Brownian bridges on complete and connected Lie groups. We show how this simulation scheme leads to absolute continuity of the Brownian bridge measure with respect to the guided process measure. This result generalizes the Euclidean result of Delyon and Hu to Lie groups. We present numerical results of the guided process in the Lie group \(\mathrm {SO}(3)\). In particular, we apply importance sampling to estimate the metric on \(\mathrm {SO}(3)\) using an iterative maximum likelihood method.
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References
Barden, D., Le, H.: Some consequences of the nature of the distance function on the cut locus in a Riemannian manifold. J. LMS 56(2), 369–383 (1997)
Delyon, B., Hu, Y.: Simulation of conditioned diffusion and application to parameter estimation. Stochast. Processes Appl. 116(11), 1660–1675 (2006)
Hsu, E.P.: Stochastic Analysis on Manifolds, vol. 38. AMS, Providence (2002)
Kendall, W.S.: The radial part of Brownian motion on a manifold: a semimartingale property. Ann. Probab. 15(4), 1491–1500 (1987)
Le, H., Barden, D.: Itô correction terms for the radial parts of semimartingales on manifolds. Probab. Theory Relat. Fields 101(1), 133–146 (1995). https://doi.org/10.1007/BF01192198
Liao, M.: Lévy Processes in Lie Groups, vol. 162. Cambridge University Press, Cambridge (2004)
Papaspiliopoulos, O., Roberts, G.: Importance sampling techniques for estimation of diffusion models. In: Statistical Methods for Stochastic Differential Equations (2012)
Shigekawa, I.: Transformations of the Brownian motion on a riemannian symmetric space. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 65, 493–522 (1984). https://doi.org/10.1007/BF00531836
Thompson, J.: Submanifold bridge processes. Ph.D. thesis, University of Warwick (2015)
Thompson, J.: Brownian bridges to submanifolds. Potential Anal. 49(4), 555–581 (2018). https://doi.org/10.1007/s11118-017-9667-1
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Jensen, M.H., Joshi, S., Sommer, S. (2021). Bridge Simulation and Metric Estimation on Lie Groups. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_47
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DOI: https://doi.org/10.1007/978-3-030-80209-7_47
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