Wick integrals

Abstract

We introduce the Wick integral ${\int }_s^{t}f(X_u)◊\mathrm{d}{X}_u$ for a class of stochastic processes $X$ of bounded $2>p$-variation which are not necessarily Gaussian. The integral is defined for a class of entire functions $f$ depending on the process. In the case of $1/2<H$-fractional Brownian motion, the Wick integral agrees with the divergence operator in Malliavin calculus. It satisfies a correction formula with the Young integral $\int f(X)\mathrm{d}X$ and an Itô formula which have infinitely many correction terms, given by integration against the cumulant functions of $X$, and reduce to familiar identities in the Gaussian case. These results are obtained by developing diagram formulae for Appell polynomials w.r.t.\ a linear span of reference random variables and extending them to series via absolute convergence in $L^1$. Our theory applies to processes taking values in the second Wiener chaos, such as the Rosenblatt process.

Substantial extension of first draft, from polynomial integrands to entire functions of exponential type