Filtration shrinkage, strict local martingales and the Föllmer measure

Details Filtration shrinkage, strict local martingales and the Föllmer measure Filtration shrinkage, strict local martingales and the Föllmer measure

2011-11-30

arxiv.org/abs/1111.7218v1

Mathematics

Probability

Scientific paper

When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. In this paper we show that the loss of the local martingale property is related to the existence of a well-behaved extension of the F\"ollmer measure. The finite variation part of the projection can then be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a diffusion setting, this leads to intuitive conditions under which its paths are singular. The results are complemented with a detailed treatment of the specific case of the inverse Bessel process.

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