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<p>We establish and investigate a deep learning-based forward scheme for nonlinear and potentially high-dimensional forward-backward stochastic differential equations (FBSDEs) with jumps, inspired by [Bender and Denk, 117 (2007), <i>Stoch. Process. Their Appl.</i>, pp.1793-1812]. The developed framework is built not upon the original FBSDEs with jumps, but rather on standard FBSDEs after suppressing the random jumps (while retaining their compensator) and appropriately adjusting the drift coefficient and driver. The associated forward scheme is built upon a recursive representation that decouples jumps at every step and converges exponentially fast to the original nonlinear FBSDE with jumps, often requiring only a few iterations to achieve sufficient accuracy. The established framework also holds novelty in its deep learning-based implementation of a wide class of forward schemes for nonlinear FBSDEs, notably whether with or without jumps. We provide a wide range of numerical results, showcasing the effectiveness of the proposed recursion and its corresponding forward scheme in approximating high-dimensional nonlinear FBSDEs with jumps without directly handling the random jumps.</p>

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A deep learning-based forward scheme for forward-backward SDEs with jumps

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  • 出版模式：

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  • 发表日期：

    2026年2月9日

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  • Reiichiro Kawai,
  • Riu Naito,
  • Toshihiro Yamada

摘要

We establish and investigate a deep learning-based forward scheme for nonlinear and potentially high-dimensional forward-backward stochastic differential equations (FBSDEs) with jumps, inspired by [Bender and Denk, 117 (2007), Stoch. Process. Their Appl., pp.1793-1812]. The developed framework is built not upon the original FBSDEs with jumps, but rather on standard FBSDEs after suppressing the random jumps (while retaining their compensator) and appropriately adjusting the drift coefficient and driver. The associated forward scheme is built upon a recursive representation that decouples jumps at every step and converges exponentially fast to the original nonlinear FBSDE with jumps, often requiring only a few iterations to achieve sufficient accuracy. The established framework also holds novelty in its deep learning-based implementation of a wide class of forward schemes for nonlinear FBSDEs, notably whether with or without jumps. We provide a wide range of numerical results, showcasing the effectiveness of the proposed recursion and its corresponding forward scheme in approximating high-dimensional nonlinear FBSDEs with jumps without directly handling the random jumps.