Consider a sequence of cadlag processes \(\{X^n\}_n\) , and some fixed function f. If f is continuous then under several modes of convergence \(X^n\rightarrow X\) implies corresponding convergence of \(f(X^n)\rightarrow f(X)\) , due to continuous mapping. We study conditions (on f, \(\{X^n\}_n\) and X) under which convergence of \(X^n\rightarrow X\) implies \(\left[ f(X^n)-f(X)\right] \rightarrow 0\) . While interesting in its own right, this also directly relates (through integration by parts and the Kunita–Watanabe inequality) to convergence of integrators in the sense \(\int _0^t Y_{s-}df(X^n_s)\rightarrow \int _0^t Y_{s-}df(X_s)\) . We show stability when \(f\in C^1\) , \(\{X^n\}_n,X\) are Dirichlet processes defined as in Coquet et al. (J Theor Probab 16:197, 2023) \(X^n\xrightarrow {a.s.}X\) , \([X^n-X]\xrightarrow {a.s.}0\) and \(\{(X^n)^*_t\}_n\) is bounded in probability. We also relax the conditions on f to being the primitive function of a cadlag function but with the additional assumption on X and that the continuous and discontinuous parts of X are independent stochastic processes (this assumption is not imposed on \(\{X^n\}_n\) however). For this setting we prove a new Itô decomposition that is a refinement of the one found in Coquet et al. (2023).
