Unifying the Dynkin and Lebesgue–Stieltjes formulae

Offer Kella, Marc Yor
2017 Journal of Applied Probability  
We establish a local martingale M associate with f(X,Y) under some restrictions on f, where Y is a process of bounded variation (on compact intervals) and either X is a jump diffusion (a special case being a Lévy process) or X is some general (càdlàg metric-space valued) Markov process. In the latter case, f is restricted to the form f(x,y)=∑ k=1 K ξ k (x)η k (y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable)
more » ... formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes an L 2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both in L 2 and almost sure is also considered and sufficient conditions for functions for which this happens are identified.
doi:10.1017/jpr.2016.98 fatcat:ois4nlyncrecvbwphs7wclc32i