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Filtering with observations on a Riemannian symmetric space. (English) Zbl 0648.93061
This paper considers the filtering problem for a stochastic system (X,Y) with X a Markov process and Y taking values in a symmetric space M. Namely, $$Y_ t=g_ t(X)W_ t$$, where $$W_ t$$ is a Brownian notion on M, independent of X, and $$g_ t$$ is an isometry of M, which is absolutely continuous in t.
The stochastic model is built by the reference probability method, using I. Shigekawa’s [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 493-533 (1984; Zbl 0518.60087)] results. The filter of X given Y is shown to be the unique solution of a stochastic differential equation w.r.t. Y in the reference probability space. For uniqueness the authors use the Kurtz- Ocone’s filtered martingale problem approach [see, T. G. Kurtz and D. L. Ocone, Ann. Probab. 16, No.1, 80-107 (1988)].
Two examples are then examined by means of the above theory. In the first one M is $$R^ d$$ and Y is a space-displacement of a Brownian notion; in the second M is the sphere $$S^ 2$$ in $$R^ 3$$ so that, even if Y can be described by a three-dimensional diffusion, being this degenerate classical filtering methods don’t work.
Reviewer: M.Piccioni

##### MSC:
 93E11 Filtering in stochastic control theory 60G35 Signal detection and filtering (aspects of stochastic processes) 60J60 Diffusion processes 58J65 Diffusion processes and stochastic analysis on manifolds 62M20 Inference from stochastic processes and prediction 53C35 Differential geometry of symmetric spaces
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