We introduce a Skorokhod type integral and prove an Itô formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Itô formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Itô integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alòs et al. (Ann. Probab. 29 (2001) 766–801).
"Itô’s formula for Gaussian processes with stochastic discontinuities." Ann. Probab. 48 (1) 458 - 492, January 2020. https://doi.org/10.1214/19-AOP1369