On the reduction of a multidimensional continuous martingale to a Brownian motion

Abstract

Knight’s well-known theorem says that orthogonal continuous local martingales, when time-changed by their brackets, become independent Brownian motions (see [1], [7]–[11]). What can be said when the given local martingales are not orthogonal? The standard way to deal with this case is to orthogonalize them, for instance with the Gram–Schmidt algorithm. This is indeed what was done by Knight himself when first using his theorem (see [9], Theorem 2.2); but he was working in a particular setting (Hunt processes) and did not give explicit formulas. Other examples where this orthogonalization is used are references [3] and [12].