Print Email Facebook Twitter The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions Title The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions Author Négyesi, B. (TU Delft Numerical Analysis) Andersson, Kristoffer (Centrum Wiskunde & Informatica (CWI)) Oosterlee, Cornelis W. (Universiteit Utrecht) Date 2024 Abstract A novel discretization is presented for decoupled forward–backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by the corresponding linear BSDE driving the trajectories of the Malliavin derivatives of the solution pair, which implies the need to provide accurate Γ estimates. The approximation is based on a merged formulation given by the Feynman–Kac formulae and the Malliavin chain rule. The continuous time dynamics is discretized with a theta-scheme. In order to allow for an efficient numerical solution of the arising semidiscrete conditional expectations in possibly high dimensions, it is fundamental that the chosen approach admits to differentiable estimates. Two fully-implementable schemes are considered: the BCOS method as a reference in the one-dimensional framework and neural network Monte Carlo regressions in case of high-dimensional problems, similarly to the recently emerging class of Deep BSDE methods (Han et al. (2018 Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510); Huré et al. (2020 Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comp., 89, 1547–1579)). An error analysis is carried out to show L2 convergence of order, under standard Lipschitz assumptions and additive noise in the forward diffusion. Numerical experiments are provided for a range of different semilinear equations up to dimensions, demonstrating that the proposed scheme yields a significant improvement in the control estimations. Subject backward stochastic differential equationsMalliavin calculusdeep BSDEneural networksBCOSgamma estimates To reference this document use: http://resolver.tudelft.nl/uuid:d6fbe041-5638-4d9c-b751-c067c10add47 DOI https://doi.org/10.1093/imanum/drad092 Embargo date 2024-08-26 ISSN 1464-3642 Source IMA Journal of Numerical Analysis Bibliographical note Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. Part of collection Institutional Repository Document type journal article Rights © 2024 B. Négyesi, Kristoffer Andersson, Cornelis W. Oosterlee Files file embargo until 2024-08-26