The paper investigates the reverse time differentiation of a stochastic exponential that occurs in smoothing, when the signal is a finite state Markov process and the observation process is a diffusion.
"Reverse Time Differentiation and Smoothing Formulae for a Finite State Markov Process." Ann. Probab. 14 (2) 480 - 489, April, 1986. https://doi.org/10.1214/aop/1176992527