Unifying the Dynkin and Lebesgue-Stieltjes formulae

We establish a local martingale M associate with (X,Y) where X is a sufficiently nice Markov process and Y is a process of bounded variation (on compact intervals). This martingale generalizes both Dynkin’s formula for Markov processes and the Lebesgue-Stieltjes integration (change of variable) formula for (right continuous) functions of bounded variation. When further, relatively easily verifiable conditions are assumed then this local Martingale becomes an L2 martingale. Convergence of the product of this Martingale with some deterministic function (of time) to zero both in L2 and a.s. is also considered and sufficient conditions for functions for which this happens are identified