Rough backward SDEs with discontinuous Young drivers
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| author | Becherer, Dirk Sun, Yuchen |
| author_facet | Becherer, Dirk Sun, Yuchen |
| contents | We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rough backward SDEs with discontinuous Young drivers Becherer, Dirk Sun, Yuchen Probability 60L90, 60J76, 60H20, 60H15, 37H30 We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$. |
| title | Rough backward SDEs with discontinuous Young drivers |
| topic | Probability 60L90, 60J76, 60H20, 60H15, 37H30 |
| url | https://arxiv.org/abs/2505.20437 |