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Path-dependent deep Galerkin method: a neural network approach to solve path-dependent partial differential equations. (English) Zbl 1471.91621

Summary: In this paper, we propose a novel numerical method for path-dependent partial differential equations (PPDEs). These equations first appeared in the seminal work of [B. Dupire, Quant. Finance, 2019 (2009), pp. 721-729], where the functional Itô calculus was developed to deal with path-dependent financial derivatives. More specifically, we generalize the deep Galerkin method (DGM) of [J. Sirignano and K. Spiliopoulos, J. Comput. Phys., 375 (2018), pp. 1339-1364] to deal with these equations. The method, which we call path-dependent DGM, consists of using a combination of feed-forward and long short-term memory architectures to model the solution of the PPDE. We then analyze several numerical examples, many from the financial mathematics literature, that show the capabilities of the method under very different situations.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Software:

DGM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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