Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of non-negative c\`adl\`ag functions  . In particular, this  implies the existence of quadratic variation in the sense of F\"ollmer quasi surely under all martingale measures. Based on the robust existence of quadratic variation and a certain topology which is induced by Vovk's outer measure, model-free It\^o integration is developed on the space of continuous paths, of non-negative c\`adl\`ag paths and of c\`adl\`ag paths with mildly restricted jumps .
Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of c\`adl\`ag functions possessing a mild restriction on the jumps directed downwards . In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative c\`adl\`ag paths and implies the existence of quadratic variation in the sense of F\"ollmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free It\^o integration is developed .
D
non-negative
A
possessing a mild restriction on the jumps directed downwards
A
result includes the existence of quadratic variation of "typical price paths" in the space of non-negative c\`adl\`ag paths and
R
quadratic variation and a certain topology which is induced by Vovk's outer measure,
the quadratic variation, a
D
on the space of continuous paths, of non-negative c\`adl\`ag paths and of c\`adl\`ag paths with mildly restricted jumps