The following question seems intuitively true to me but I would appreciate if someone could provide a through/rigorous explanation:
Let f(t) and g(t) be continuous-time processes defined on the same sample space: Is the filtration generated by (f(t),g(t)) the same as that generated by (f(t),f(t)+g(t))?
More generally, when will the filtration generated by a function of a stochastic process be the same as the one generated by the original process?
Many thanks