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Stopping non-commutative processes. (English) Zbl 0585.60058

The paper discusses stopping times and stopping in the context of the Clifford probability gauge space. The authors show that \(L^ 2\)- martingales can be stopped and they demonstrate that the formalism is very similar to the commutative case. However they remain silent upon the question of whether or not one can stop processes other than \(L^ 2\)- martingales. Apart from the optional stopping theorem there is also a random time martingale convergence theorem. It would be interesting to know whether an ’almost everywhere’ version of this result holds.
The example of a ”time” given in the final section of the paper should not be taken too seriously, the formalism developed in this paper shows only that adapted spectral families are time-like not that all adapted families are times.

MSC:

60H05 Stochastic integrals
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
60G44 Martingales with continuous parameter
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References:

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[2] DOI: 10.1016/0022-1236(82)90066-0 · Zbl 0492.46051 · doi:10.1016/0022-1236(82)90066-0
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