In this paper, we give a pathwise development of stochastic integrals with respect to iterated Brownian motion. We also provide a detailed analysis of the variations of iterated Brownian motion. These variations are linked to Brownian motion in random scenery and iterated Brownian motion itself.
"Stochastic calculus for Brownian motion on a Brownian fracture." Ann. Appl. Probab. 9 (3) 629 - 667, August 1999. https://doi.org/10.1214/aoap/1029962807