Nonlinear Stochastic Filtering with Volterra Gaussian noises
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915373089030144 |
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| author | Cass, Thomas Crisan, Dan Iannucci, Andrea |
| author_facet | Cass, Thomas Crisan, Dan Iannucci, Andrea |
| contents | We consider a nonlinear filtering problem for a signal-observation system driven by a Volterra-type Gaussian rough path, whose sample paths may exhibit greater roughness than those of Brownian motion. The observation process includes a Volterra-type drift, introducing both memory effects and low regularity in the dynamics. We prove well-posedness of the associated rough differential equations and the Kallianpur-Striebel. We then establish robustenss properties of the filter and study the existence, smoothness, and time regularity of its density using partial Malliavin calculus. Finally, we show that, in the one-dimensional case, the density of the unnormalized filter solves a rough partial differential equation, providing a rough-path analogue of the Zakai equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09637 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Stochastic Filtering with Volterra Gaussian noises Cass, Thomas Crisan, Dan Iannucci, Andrea Probability We consider a nonlinear filtering problem for a signal-observation system driven by a Volterra-type Gaussian rough path, whose sample paths may exhibit greater roughness than those of Brownian motion. The observation process includes a Volterra-type drift, introducing both memory effects and low regularity in the dynamics. We prove well-posedness of the associated rough differential equations and the Kallianpur-Striebel. We then establish robustenss properties of the filter and study the existence, smoothness, and time regularity of its density using partial Malliavin calculus. Finally, we show that, in the one-dimensional case, the density of the unnormalized filter solves a rough partial differential equation, providing a rough-path analogue of the Zakai equation. |
| title | Nonlinear Stochastic Filtering with Volterra Gaussian noises |
| topic | Probability |
| url | https://arxiv.org/abs/2506.09637 |