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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.08497 |
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| _version_ | 1866915059139084288 |
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| author | Pellat, Rhoss Likibi Fonka, Emmanuel Che Pamen, Olivier Menoukeu |
| author_facet | Pellat, Rhoss Likibi Fonka, Emmanuel Che Pamen, Olivier Menoukeu |
| contents | We investigate the convergence rate for the time discretization of a class of quadratic backward SDEs -- potentially involving path-dependent terminal values -- when coupled with non-standard Lipschitz-type forward SDEs. In our review of the explicit time-discretization schemes in the spirit of Pagès \& Sagna (see \cite{PaSa18}), we achieve an error control close to $\frac{1}{2}$, even under the modest assumptions considered in this work (see \cite{ChaRichou16}, for comparison). A central element of our approach is a thorough re-examination of Zhang's $L^2\text{-time regularity}$ of the martingale integrand $Z$ which follows from an extension of the first-order variational regularity for this class of singular forward-backward SDEs with non-uniform Cauchy-Lipschitz drivers. This is complemented by the recently introduced caracterisation of stochastic processes of {\it bounded mean oscillation} (abbreviated as $\bmo$) by K. Lê (see \cite{Le22}) which we used to derive an $L^p\text{-version}$ of the strong approximation of SDEs with singular drifts from Dareiotis \& Gerencsér (see \cite{DaGe20}). As such, this study addresses a crucial gap in the numerical analysis of forward-backward SDEs (FBSDEs). To our knowledge, for the first time, the impact of regularization by noise on Euler-Maruyama numerical schemes for singular forward SDEs has been successfully transferred to enhance the convergence rate of the discrete time approximations for solutions to backward SDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Time discretization of Quadratic Forward-Backward SDEs with singular drifts Pellat, Rhoss Likibi Fonka, Emmanuel Che Pamen, Olivier Menoukeu Probability We investigate the convergence rate for the time discretization of a class of quadratic backward SDEs -- potentially involving path-dependent terminal values -- when coupled with non-standard Lipschitz-type forward SDEs. In our review of the explicit time-discretization schemes in the spirit of Pagès \& Sagna (see \cite{PaSa18}), we achieve an error control close to $\frac{1}{2}$, even under the modest assumptions considered in this work (see \cite{ChaRichou16}, for comparison). A central element of our approach is a thorough re-examination of Zhang's $L^2\text{-time regularity}$ of the martingale integrand $Z$ which follows from an extension of the first-order variational regularity for this class of singular forward-backward SDEs with non-uniform Cauchy-Lipschitz drivers. This is complemented by the recently introduced caracterisation of stochastic processes of {\it bounded mean oscillation} (abbreviated as $\bmo$) by K. Lê (see \cite{Le22}) which we used to derive an $L^p\text{-version}$ of the strong approximation of SDEs with singular drifts from Dareiotis \& Gerencsér (see \cite{DaGe20}). As such, this study addresses a crucial gap in the numerical analysis of forward-backward SDEs (FBSDEs). To our knowledge, for the first time, the impact of regularization by noise on Euler-Maruyama numerical schemes for singular forward SDEs has been successfully transferred to enhance the convergence rate of the discrete time approximations for solutions to backward SDEs. |
| title | Time discretization of Quadratic Forward-Backward SDEs with singular drifts |
| topic | Probability |
| url | https://arxiv.org/abs/2412.08497 |