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On pathwise quadratic variation for càdlàg functions. (English) Zbl 1406.60082

Summary: We revisit Föllmer’s concept of quadratic variation of a càdlàg function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of càdlàg processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition which implies the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.

MSC:

60H05 Stochastic integrals
26B35 Special properties of functions of several variables, Hölder conditions, etc.
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