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Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

Author

Listed:
  • Riu Naito

    (Hitotsubashi University
    Japan Post Insurance)

  • Toshihiro Yamada

    (Hitotsubashi University
    Japan Science and Technology Agency (JST))

Abstract

The paper introduces a new deep learning-based high-order spatial approximation for a solution of a high-dimensional Kolmogorov equation where the initial condition is only assumed to be a continuous function and the condition on the vector fields associated with the differential operator is very general, i.e. weaker than Hörmander’s hypoelliptic condition. In particular, the deep learning-based method is constructed based on the Kusuoka approximation. Numerical results for high-dimensional partial differential equations up to 500-dimension cases appearing in option pricing problems show the validity of the method. As an application, a computation scheme for the delta is shown using “deep” numerical differentiation.

Suggested Citation

  • Riu Naito & Toshihiro Yamada, 2024. "Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance," Computational Economics, Springer;Society for Computational Economics, vol. 64(3), pages 1443-1461, September.
    • Handle: RePEc:kap:compec:v:64:y:2024:i:3:d:10.1007_s10614-023-10476-2
    DOI: 10.1007/s10614-023-10476-2
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