Functional Data Analysis with Rough Sampled Paths?

Functional data are typically modeled as sampled paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The required smoothness allows for the use of smoothing techniques but excludes many stochastic processes, most notably diffusion processes. Such processes would otherwise be well within the realm of functional data analysis, at least under complete observation. In this short note we demonstrate that a simple modification of existing methods allows for the functional data analysis of processes with nowhere differentiable sample paths, even when these are discretely and noisily observed, including under irregular and sparse designs. By way of simulation it is shown that this is not a theoretical curiosity, but can work well in practice, hinting at potential closer links with the field of diffusion inference.