On a class of generalized Takagi functions with linear pathwise quadratic variation

2015/01/05 by Alexander Schied, Schied, Alexander
Economics, Econometrics and Finance · Mathematics · #26A15 #26A30 #26A45 #60H05 #Classical Analysis and ODEs (math.CA) #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR) #math.CA #math.PR #msc:26A15 #msc:26A30 #msc:26A45 #msc:60H05 #q-fin.MF

paper · pdf · doi:10.48550/arxiv.1501.00837

arxiv created 2015/08/13 · arxiv updated 2015/08/14

Abstract

We consider a class \mathscrX of continuous functions on [0,1] that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in \mathscrX admits a linear pathwise quadratic variation and can thus serve as an integrator in Föllmer's pathwise Itō calculus. We derive several uniform properties of the class \mathscrX. For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in \mathscrX. Furthermore, we give an example of a pair x,y∈\mathscrX such that the quadratic variation of the sum x+y does not exist.

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