Ito's Formula for Gaussian Processes with Stochastic Discontinuities

TL;DR

This paper develops an Ito formula for a broad class of Gaussian processes with stochastic discontinuities, unifying existing formulas for continuous and discontinuous cases using a Skorokhod integral.

Contribution

It introduces a Skorokhod type integral and proves an Ito formula applicable to Gaussian processes with stochastic discontinuities, extending classical results.

Findings

Unified Ito formula for Gaussian processes with discontinuities

Extension of classical Ito formulas to non-martingale Gaussian processes

Framework for stochastic calculus with discontinuous Gaussian processes

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).