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RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-Itô integrals
Chunhao Cai, Cong Zhang · 2024-10-24 · via math.PR updates on arXiv.org

Fractional Brownian motions(fBMs) are not semimartingales so the classical theory of Itô integral can't apply to fBMs. Wick integration as one of the applications of Malliavin calculus to stochastic analysis is a fine definition for fBMs. We consider the fractional forward backward stochastic differential equations(fFBSDEs) driven by a fBM that have the Hurst parameter in (1/2,1) where $\int_{0}^{t} f_s \, dB_s^H$ is in the sense of a Wick integral, and relate our fFBSDEs to the system of partial differential equations by using an analogue of the Itô formula for Wick integrals. And we develop a deep learning algorithm referred to as the RNN-BSDE method based on recurrent neural networks which is exactly designed for solving high-dimensional fractional BSDEs and their corresponding partial differential equations.

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