RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-It\^o integrals

TL;DR

This paper introduces a novel deep learning RNN-based method for solving high-dimensional fractional backward stochastic differential equations driven by fractional Brownian motion with Wick integrals, connecting stochastic analysis and PDEs.

Contribution

It develops the RNN-BSDE method tailored for high-dimensional fractional BSDEs with Wick integrals, bridging stochastic calculus and deep learning techniques.

Findings

The RNN-BSDE method effectively solves high-dimensional fractional BSDEs.

The approach connects fractional stochastic equations to PDEs via Wick integrals.

Demonstrates the applicability of deep learning in fractional stochastic analysis.

Abstract

Fractional Brownian motions(fBMs) are not semimartingales so the classical theory of It\^o 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 It\^o 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.