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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.14986 |
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| _version_ | 1866911323965620224 |
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| author | Bellingeri, Carlo Ferrucci, Emilio |
| author_facet | Bellingeri, Carlo Ferrucci, Emilio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14986 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wick integrals Bellingeri, Carlo Ferrucci, Emilio Probability 60H05, 60G22, 60C05 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. |
| title | Wick integrals |
| topic | Probability 60H05, 60G22, 60C05 |
| url | https://arxiv.org/abs/2512.14986 |