Poisson imbedding meets the Clark-Ocone formula
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909166672543744 |
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| author | Hillairet, Caroline Peyrat, Thomas Réveillac, Anthony |
| author_facet | Hillairet, Caroline Peyrat, Thomas Réveillac, Anthony |
| contents | In this paper we develop a representation formula of Clark-Ocone type for any integrable Poisson functionals, which extends the Poisson imbedding for point processes. This representation formula differs from the classical Clark-Ocone formula on three accounts. First the representation holds with respect to the Poisson measure instead of the compensated one; second the representation holds true in L1 and not in L2; and finally contrary to the classical Clark-Ocone formula the integrand is defined as a pathwise operator and not as a L2-limiting object. We make use of Malliavin's calculus and of the pseudo-chaotic decomposition with uncompensated iteraded integrals to establish this Pseudo-Clark-Ocone representation formula and to characterize the integrand, which turns out to be a predictable integrable process. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poisson imbedding meets the Clark-Ocone formula Hillairet, Caroline Peyrat, Thomas Réveillac, Anthony Probability 60G55, 60G57, 60H07 In this paper we develop a representation formula of Clark-Ocone type for any integrable Poisson functionals, which extends the Poisson imbedding for point processes. This representation formula differs from the classical Clark-Ocone formula on three accounts. First the representation holds with respect to the Poisson measure instead of the compensated one; second the representation holds true in L1 and not in L2; and finally contrary to the classical Clark-Ocone formula the integrand is defined as a pathwise operator and not as a L2-limiting object. We make use of Malliavin's calculus and of the pseudo-chaotic decomposition with uncompensated iteraded integrals to establish this Pseudo-Clark-Ocone representation formula and to characterize the integrand, which turns out to be a predictable integrable process. |
| title | Poisson imbedding meets the Clark-Ocone formula |
| topic | Probability 60G55, 60G57, 60H07 |
| url | https://arxiv.org/abs/2404.07541 |