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Generalization of Doob’s optional sampling theorem for deformed submartingales. (English. Russian original) Zbl 1326.60054

Russ. Math. Surv. 68, No. 6, 1139-1141 (2013); translation from Usp. Mat. Nauk 68, No. 6, 175-176 (2013).
Given a measure space with an increasing family of \(\sigma\)-algebras \((F_n)_{n \geq 1}\), assume that for each point of time \(n\) there is given a probability measure \(Q_n\) for the \(\sigma\)-algebra \(F_n\) at time \(n\). Then this family of probability measures is called a deformation of the first type if the probability measure on \(F_n\) induced by \(Q_{n+1}\) is absolutely continuous with respect to \(Q_n\).
Reminiscent of a submartingale, a process \(Z\) is called a deformed submartingale of the first type if \(Z_n \leq E_{Q_{n+1}}(Z_{n+1}|F_n)\) for all \(n\).
The main results are certain stopping time results for deformed submartingales reminiscent of classical cases like Doob’s optional sampling theorem.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G48 Generalizations of martingales
60G42 Martingales with discrete parameter
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