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Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations

Lorenc Kapllani, Long Teng

TL;DR

This work tackles solving high-dimensional nonlinear BSDEs by reframing the problem as a global optimization with local losses that include the terminal condition. The authors introduce LaDBSDE, a forward deep learning scheme that learns $Y$ via a neural network and $Z$ via automatic differentiation, using an iterative Euler discretization to enforce dynamics across time steps. By aggregating local losses that reach the terminal state, LaDBSDE mitigates poor local minima and provides accurate approximations for the entire time domain, outperforming existing forward schemes (DBSDE and LDBSDE) in numerical experiments, including high-dimensional pricing problems. The approach blends neural network function approximation with gradient-based optimization and a carefully designed loss structure, offering a scalable tool for nonlinear BSDEs with many dimensions and complex drivers.

Abstract

In this work, we propose a new deep learning-based scheme for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization, where the local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler discretization of the time integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.

Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations

TL;DR

This work tackles solving high-dimensional nonlinear BSDEs by reframing the problem as a global optimization with local losses that include the terminal condition. The authors introduce LaDBSDE, a forward deep learning scheme that learns via a neural network and via automatic differentiation, using an iterative Euler discretization to enforce dynamics across time steps. By aggregating local losses that reach the terminal state, LaDBSDE mitigates poor local minima and provides accurate approximations for the entire time domain, outperforming existing forward schemes (DBSDE and LDBSDE) in numerical experiments, including high-dimensional pricing problems. The approach blends neural network function approximation with gradient-based optimization and a carefully designed loss structure, offering a scalable tool for nonlinear BSDEs with many dimensions and complex drivers.

Abstract

In this work, we propose a new deep learning-based scheme for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization, where the local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler discretization of the time integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.

Paper Structure

This paper contains 12 sections, 35 equations, 16 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The nonlinear Feynman-Kac formula
  4. Neural Networks as function approximators
  5. Learning long-term dependencies in recurrent neural networks
  6. Forward time discretization of BSDEs
  7. The forward deep learning-based schemes for BSDEs
  8. The deep BSDE scheme weinan2017deep
  9. The local deep BSDE scheme raissi2018forward
  10. The locally additive deep BSDE scheme
  11. Numerical results
  12. Conclusion

Figures (16)

  • Figure 1: The architecture of the LaDBSDE scheme.
  • Figure 2: Realizations of $5$ independent paths for Example \ref{['ex1']} using $d = 1$ and $N=240$. $(Y_t, Z_t)$ and $(\mathcal{Y}_t^{\theta},\mathcal{Z}_t^{\theta})$ are exact and learned solutions for $t \in [0, T]$, respectively.
  • Figure 3: The mean regression errors $(\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i})$ at time step $t_i, i = 0, \cdots, N-1$ for Example \ref{['ex1']} using $d=1$ and $N=240$. The standard deviation is given in the shaded area.
  • Figure 4: Realizations of $5$ independent paths for Example \ref{['ex1']} using $d = 100$ and $N=120$. $(Y_t, Z_t^1)$ and $(\mathcal{Y}_t^{\theta},\mathcal{Z}_t^{1,\theta})$ are exact and learned solutions for $t \in [0, T]$, respectively.
  • Figure 5: The mean regression errors $(\bar{\epsilon}_{Y_i}, \bar{\epsilon}_{Z_i})$ at time step $t_i, i = 0, \cdots, N-1$ for Example \ref{['ex1']} using $d=100$ and $N=120$. The standard deviation is given in the shaded area.
  • ...and 11 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Example 1
  • Example 2
  • Example 3
  • Example 4