Wick integrals

TL;DR

The paper develops a pathwise Wick integration theory for non-Gaussian processes in the Young regime by building an associative Wick product via Appell polynomials and diagrammatic calculus. It establishes convergence of Wick-integral sums to a Young integral with cumulant-based correction terms and delivers Itô-type change-of-variables formulas whose correction structure generalizes the Gaussian case. The framework is extended to processes living in finite Wiener chaos, with a detailed treatment of density/non-degeneracy and explicit results for the Rosenblatt process, including a comparison to existing approaches. Collectively, the work provides a versatile non-Gaussian stochastic calculus that unifies diagrammatic, cumulant, and chaos-expansion methods.

Abstract

We introduce the Wick integral $\int_s^t p(X_u) \Diamond \mathrm{d} X_u$ for a class of stochastic processes $X$ which are not necessarily Gaussian, in the regime of bounded $2> q$-variation. The integral is defined for polynomial integrands, and has the property of being centred if $X$ is such. 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 $\int p(X)\mathrm{d} X$ and an Itô formula which have arbitrarily many correction terms (only limited by the degree of $p$), given by integration against the cumulant functions of $X$, and reduce to familiar identities in the Gaussian case. These results are obtained by first developing diagram formulae for Appell polynomials. Our theory applies to a range of processes taking values in bounded Wiener chaos, such as the Rosenblatt process.

Figures (3)

• Figure 1: Examples of diagrams. The first and third diagrams are non-flat, while the second has a flat edge in the first row. The first two diagrams are connected (the second because of the residual set), but not the last.
• Figure 2: Drawing representing the above partitions. The $\tau$ partition (in blue) is flat, while generally only part of the sets in $\sigma$ (in red) are flat. In the next part of the proof, we will group $\tau$ with the sets in $\sigma$ which are flat.
• Figure 3: The diagram $F$, rotated by 90$^\circ$ counterclockwise to improve the layout, so that rows have become columns. The dividing lines represent connected components, with the blue terms those coming from $\rho$ and the red terms coming from $R$. The proof consists of considering together connected components without a residual set, in the picture the first three starting from the right. In the proof, edges in the diagram with a single node per row $(\rho, R) \in \mathcal{D}([m])$ correspond to connected components in $F$.

Theorems & Definitions (44)

• proposition 1.1: Appell polynomials associated to a measure
• remark 1.2
• remark 1.3: Lack of associativity
• remark 1.4: Invariance under linear transformations
• remark 1.5: Well defined product
• example 1.6: Poisson
• definition 1.7
• remark 1.8
• proposition 1.9: Giraitis_1986
• theorem 1.10: Product formula