Parameter dependent rough SDEs with applications to rough PDEs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909700414504960 |
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| author | Bugini, Fabio Friz, Peter K. Stannat, Wilhelm |
| author_facet | Bugini, Fabio Friz, Peter K. Stannat, Wilhelm |
| contents | Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and Lê in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise filtering and control, volatility modelling in finance and mean-field stochastic dynamics with common noise ...). While the full dynamics may be highly non-Markovian, the conditional dynamics often are. In natural (and even linear) situations, the resulting stochastic PDEs can be beyond existing technology. The present work then tackles a key problem in this context, which is the well-posedness of regular solution to the rough Kolmogorov backward equation. To this end, we study parameter dependent rough SDEs in sense of $\mathscr{L}$-differentiability (as in Krylov, 2008). In companion works, we will show how this removes dimension-dependent regularity assumptions for well-posedness of the Zakai, Kushner-Stratonovich and nonlinear Fokker-Planck stochastic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parameter dependent rough SDEs with applications to rough PDEs Bugini, Fabio Friz, Peter K. Stannat, Wilhelm Probability 60H10, 60H15, 60L20, 60L50 Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and Lê in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise filtering and control, volatility modelling in finance and mean-field stochastic dynamics with common noise ...). While the full dynamics may be highly non-Markovian, the conditional dynamics often are. In natural (and even linear) situations, the resulting stochastic PDEs can be beyond existing technology. The present work then tackles a key problem in this context, which is the well-posedness of regular solution to the rough Kolmogorov backward equation. To this end, we study parameter dependent rough SDEs in sense of $\mathscr{L}$-differentiability (as in Krylov, 2008). In companion works, we will show how this removes dimension-dependent regularity assumptions for well-posedness of the Zakai, Kushner-Stratonovich and nonlinear Fokker-Planck stochastic equations. |
| title | Parameter dependent rough SDEs with applications to rough PDEs |
| topic | Probability 60H10, 60H15, 60L20, 60L50 |
| url | https://arxiv.org/abs/2409.11330 |