Sticky processes, local and true martingales

We prove that for a so-called sticky process \(S\) there exists an equivalent probability \(Q\) and a \(Q\)-martingale \(\tilde{S}\) that is arbitrarily close to \(S\) in \(L^p(Q)\) norm. For continuous \(S\), \(\tilde{S}\) can be chosen arbitrarily close to \(S\) in supremum norm. In the case where \(S\) is a local martingale we may choose \(Q\) arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance.