On a relation between stochastic integration and geometric measure theory

Two problems are addressed for the path of certain stochastic processes: a) do they define currents? b) are these currents of a classical type? A general answer to question a) is given for processes like semimartingales or with Lyons-Zheng structure. As to question b), it is shown that H\"{o}lder continuous paths with exponent $\gamma > 1/2$ define integral flat chains.