Riemann-Skorohod and Stratonovich integrals for Gaussian processes

TL;DR

This paper explores the relationship between Skorohod and Stratonovich integrals for Gaussian processes, establishing conversion formulas under conditions on covariance variations and linking the difference to Young integrals.

Contribution

It provides new conditions under which Skorohod and Stratonovich integrals are related and characterizes their difference as a Young integral, extending the theory for Gaussian processes.

Findings

Conversion formula between Skorohod and Stratonovich integrals under finite variation conditions.

Identification of the difference as a Young integral.

Skorohod integral as a limit of Skorohod-Riemann sums.

Abstract

In this paper we consider Skorohod and Stratonovich-type integrals in a general setting of Gaussian processes. We show that a conversion formula holds when the covariance functions of the Gaussian process are of finite $\rho$-variation for $\rho\geq 1$ and that the diagonals of covariance functions are of finite $\rho'$-variation for $\rho'\geq 1$ such that $\frac{1}{\rho'}+\frac{1}{2\rho}>1$. The difference between the two types of integrals is identified with a Young integral. We also show that the Skorohod integral is the limit of a $[\rho]$-th order Skorohod-Riemann sum.