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Licensed Unlicensed Requires Authentication Published by De Gruyter July 13, 2022

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

  • Martin Hutzenthaler EMAIL logo , Arnulf Jentzen , Thomas Kruse and Tuan Anh Nguyen

Abstract

Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approximation methods for nonlinear BSDEs, until today there has been no numerical approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of nonlinear BSDEs in the sense that the number of computational operations of the numerical approximation method to approximatively compute one sample path of the BSDE solution grows at most polynomially in both the reciprocal 1/ε of the prescribed approximation accuracy ε ∈ (0, ∞) and the dimension d ∈ N = {1, 2, 3, . . .} of the BSDE. It is the key contribution of this article to overcome this obstacle by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.

JEL Classification: 65C30

Acknowledgment

We thank Benno Kuckuck for several helpful discussions. This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure and through the research grant HU1889/6-2. This project has been partially supported by the startup fund project of Shenzhen Research Institute of Big Data under grant No. T00120220001.

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Received: 2021-09-15
Revised: 2022-06-28
Accepted: 2022-07-05
Published Online: 2022-07-13
Published in Print: 2023-03-28

© 2023 Walter de Gruyter GmbH, Berlin/ Boston, Germany

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