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 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.
From: Emilio Ferrucci [view email]
[v1]
Wed, 17 Dec 2025 00:47:48 UTC (47 KB)
[v2]
Mon, 27 Jul 2026 19:34:49 UTC (75 KB)
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