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RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-Itô integrals

Chunhao Cai, Cong Zhang

TL;DR

This work tackles high-dimensional fractional backward SDEs driven by fractional Brownian motion with Wick integration ($H \in (1/2,1)$) and links them to nonlinear PDEs via a Wick-Itô calculus framework. It introduces the RNN-BSDE method, using stacked RNNs or LSTMs to approximate the $Z$-process and exploit long-memory effects for high-dimensional problems, with update rules that incorporate Wick terms and an end-to-end loss on $Y_T$. The authors establish existence/uniqueness results and provide convergence analysis for the discretized scheme, then demonstrate extensive numerical experiments on fractional Black-Scholes equations (1D and up to $d=100$), nonlinear fractional models, and a semilinear heat equation, showing that LSTM-based architectures generally outperform non-recurrent baselines and that the approach scales to high dimensions. The results indicate that the proposed method offers a practical and effective tool for solving complex fractional stochastic systems and their associated PDEs in financial and physical contexts, where memory effects and non-Markovian dynamics are essential.

Abstract

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.

RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-Itô integrals

TL;DR

This work tackles high-dimensional fractional backward SDEs driven by fractional Brownian motion with Wick integration () and links them to nonlinear PDEs via a Wick-Itô calculus framework. It introduces the RNN-BSDE method, using stacked RNNs or LSTMs to approximate the -process and exploit long-memory effects for high-dimensional problems, with update rules that incorporate Wick terms and an end-to-end loss on . The authors establish existence/uniqueness results and provide convergence analysis for the discretized scheme, then demonstrate extensive numerical experiments on fractional Black-Scholes equations (1D and up to ), nonlinear fractional models, and a semilinear heat equation, showing that LSTM-based architectures generally outperform non-recurrent baselines and that the approach scales to high dimensions. The results indicate that the proposed method offers a practical and effective tool for solving complex fractional stochastic systems and their associated PDEs in financial and physical contexts, where memory effects and non-Markovian dynamics are essential.

Abstract

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 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.

Paper Structure

This paper contains 16 sections, 14 theorems, 111 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Wick calculus
  3. Stochastic derivative
  4. Fractional Wick Itô Skorohod integrals in $L^2$
  5. Uniqueness and existence of fractional backward SDEs and systems of PDEs
  6. RNN-BSDE method
  7. Main ideas of the RNN-BSDE method
  8. More detail of the RNN-BSDE method and an example of solving fBSDEs
  9. Convergence analysis
  10. Numerical examples
  11. Fractional Black-Scholes equation
  12. Results in the one-dimension case($d=1$)
  13. Results in the high-dimension case($d=50$)
  14. Nonlinear fractional Black-Scholes equation with different interest rates for borrowing and lending($d=100$)
  15. A semilinear heat equation with variable coefficients($d = 50$)
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $F \in L^2(P_\phi)$. Then there exist constants $c_\alpha \in \mathbb{R}$, $\alpha \in \mathcal{I}$, such that Moreover, where $\alpha! = \alpha_1!\alpha_2!\cdots\alpha_m!$ if $\alpha = (\alpha_1, \dots, \alpha_m) \in \mathcal{I}$.

Figures (6)

  • Figure 1: Rough sketch of the architecture of the stacked RNN. $t_n$ is simply written as $n$, $h_{j}^{(i)}$ means the hidden state of the $i$th hidden layer at time $t_j$.
  • Figure 2: Rough sketch of the architecture of the RNN-BSDE method. $\theta$ represents the parameters to be trained.
  • Figure 3: Mean of the loss function and relative $L^1$-approximation error of $u(0,x_0)$ in the $1-d$ case of the PDE \ref{['eqn33']}. (a.1) mean of the loss function when $H = 1/2$; (a.2)relative $L^1$-approximation error when $H = 1/2$; (b.1) mean of the loss function when $H = 3/4$; (b.2)relative $L^1$-approximation error when $H = 3/4$.
  • Figure 4: Mean of the loss function and mean of the approximations of $u(0,x_0)$ in the $50-d$ case of the PDE \ref{['eqn36']}. (c.1) mean of the loss function when $H = 1/2$; (c.2)mean of the approximations of $u(0,x_0)$ when $H = 1/2$; (d.1) mean of the loss function when $H = 3/4$; (d.2)mean of the approximations of $u(0,x_0)$ when $H = 3/4$.
  • Figure 5: Mean of the loss function and relative $L^1$-approximation error of $u(0,x_0)$ in the $100-d$ case of the PDE \ref{['eqn49']}. (a) mean of the loss function ; (b)relative $L^1$-approximation error of $u(0,x_0)$ when $H = 3/4$.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition 1.1: fBMfbmdef
  • Theorem 2.1: The fractional Wiener-Itô chaos expansion
  • Definition 2.2: HU2011
  • Definition 2.3: Wick product in $(S)^*_H$
  • Definition 2.4: Fractional Wick Itô Skorohod integrals
  • Definition 2.5
  • Definition 2.6: Malliavin derivative
  • Lemma 2.7: Biagini
  • Lemma 2.8: The chain rule
  • Definition 2.9: Fractional Wick Itô Skorohod integrals in $L^2$
  • ...and 19 more