Wick integrals

Abstract

We introduce the Wick integral $\int_s^t f(X_u) \Diamond \mathrm{d} X_u$ 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 It\^o 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.