Convergence of the deep BSDE method for stochastic control problems formulated through the stochastic maximum principle

ArXiv ID: 2401.17472 “View on arXiv”

Authors: Unknown

Abstract

It is well-known that decision-making problems from stochastic control can be formulated by means of a forward-backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. 2022 proposed an efficient deep learning algorithm based on the stochastic maximum principle (SMP). In this paper, we provide a convergence result for this deep SMP-BSDE algorithm and compare its performance with other existing methods. In particular, by adopting a strategy as in Han and Long 2020, we derive a-posteriori estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. We present numerical examples for high-dimensional stochastic control problems, both in case of drift- and diffusion control, which showcase superior performance compared to existing algorithms.

Keywords: stochastic control, forward-backward stochastic differential equation (FBSDE), deep learning, stochastic maximum principle (SMP), convergence analysis, General Financial Instruments

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper is highly mathematical, featuring advanced stochastic calculus, FBSDE theory, and detailed convergence proofs with a-posteriori error estimates. It demonstrates strong empirical rigor by presenting numerical experiments on high-dimensional stochastic control problems, comparing performance against existing algorithms.

  flowchart TD
    A["Research Goal<br/>Analyze convergence of<br/>deep SMP-BSDE algorithm for<br/>high-dimensional stochastic control"] --> B["Key Inputs<br/>• FBSDE formulations (Drift/Diffusion control)<br/>• Neural network approximations<br/>• SMP-inspired loss functions"]
    B --> C["Methodology Steps<br/>1. SMP-based FBSDE formulation<br/>2. A-posteriori error estimation<br/>3. Discretization error analysis"]
    C --> D["Computational Process<br/>• Deep learning optimizer<br/>• Monte Carlo simulations<br/>• High-dimensional PDE approximation"]
    D --> E["Key Outcomes<br/>• Convergence proof with error bounds<br/>• Loss functional < A-posteriori error < Discretization error<br/>• Superior performance vs. existing methods<br/>• Numerical validation on 100+ dimensions"]