Paper: Oct 13,2024
math.NA
ID:2410.09666
A forward scheme with machine learning for forward-backward SDEs with jumps by decoupling jumps
Forward-backward stochastic differential equations (FBSDEs) have been
generalized by introducing jumps for better capturing random phenomena, while
the resulting FBSDEs are far more intricate than the standard one from every
perspective. In this work, we establish a forward scheme for potentially
high-dimensional FBSDEs with jumps, taking a similar approach to [Bender and
Denk, 117 (2007), Stoch. Process. Their Appl., pp.1793-1812], with the aid of
machine learning techniques for implementation. The developed forward scheme is
built upon a recursive representation that decouples random jumps at every step
and converges exponentially fast to the original FBSDE with jumps, often
requiring only a few iterations to achieve sufficient accuracy, along with the
error bound vanishing for lower jump intensities. The established framework
also holds novelty in its neural network-based implementation of a wide class
of forward schemes for FBSDEs, notably whether with or without jumps. We
provide an extensive collection of numerical results, showcasing the
effectiveness of the proposed recursion and its corresponding forward scheme in
approximating high-dimensional FBSDEs with jumps (up to 100-dimension) without
directly handling the random jumps.
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Paper Author: Reiichiro Kawai,Riu Naito,Toshihiro Yamada
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