Ito's Formula for Gaussian Processes with Stochastic Discontinuities

Abstract

We introduce a Skorokhod type integral and prove an Ito formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Ito formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Ito integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alos et al. (Ann. Probab. 29, 2001).