Paper: arXiv 2401.17472

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: 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.

Research Flowchart

  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"]

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0 of 38 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.