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Deep Learning for the Multiple Optimal Stopping Problem

Mathieu Laurière, Mehdi Talbi

TL;DR

The paper tackles high-dimensional multiple optimal stopping by integrating dynamic programming with neural network value-function approximations, explicitly learning the value surface rather than the stopping policy. It introduces two backward-induction algorithms: Exhaustive-DBMS (E-DBMS) that preserves the full DP but incurs exponential search cost, and Partial-DBMS (P-DBMS) that reduces cost by restricting to a single stop per step, with a trade-off in discretization error. The authors prove convergence results for both algorithms, decompose errors into neural-network approximation, sampling, and discretization components, and derive explicit diffusion-level error bounds for Euler discretizations. Numerical experiments on high-dimensional basket American options and nonlinear utilities demonstrate the approach's scalability and accuracy, validating its applicability to swing options and related problems in finance. Overall, the framework provides a rigorous, scalable method for solving complex multiple stopping problems in high dimensions with provable convergence guarantees and practical implementation guidance.

Abstract

This paper presents a novel deep learning framework for solving multiple optimal stopping problems in high dimensions. While deep learning has recently shown promise for single stopping problems, the multiple exercise case involves complex recursive dependencies that remain challenging. We address this by combining the Dynamic Programming Principle with neural network approximation of the value function. Unlike policy-search methods, our algorithm explicitly learns the value surface. We first consider the discrete-time problem and analyze neural network training error. We then turn to continuous problems and analyze the additional error due to the discretization of the underlying stochastic processes. Numerical experiments on high-dimensional American basket options and nonlinear utility maximization demonstrate that our method provides an efficient and scalable method for the multiple optimal stopping problem.

Deep Learning for the Multiple Optimal Stopping Problem

TL;DR

The paper tackles high-dimensional multiple optimal stopping by integrating dynamic programming with neural network value-function approximations, explicitly learning the value surface rather than the stopping policy. It introduces two backward-induction algorithms: Exhaustive-DBMS (E-DBMS) that preserves the full DP but incurs exponential search cost, and Partial-DBMS (P-DBMS) that reduces cost by restricting to a single stop per step, with a trade-off in discretization error. The authors prove convergence results for both algorithms, decompose errors into neural-network approximation, sampling, and discretization components, and derive explicit diffusion-level error bounds for Euler discretizations. Numerical experiments on high-dimensional basket American options and nonlinear utilities demonstrate the approach's scalability and accuracy, validating its applicability to swing options and related problems in finance. Overall, the framework provides a rigorous, scalable method for solving complex multiple stopping problems in high dimensions with provable convergence guarantees and practical implementation guidance.

Abstract

This paper presents a novel deep learning framework for solving multiple optimal stopping problems in high dimensions. While deep learning has recently shown promise for single stopping problems, the multiple exercise case involves complex recursive dependencies that remain challenging. We address this by combining the Dynamic Programming Principle with neural network approximation of the value function. Unlike policy-search methods, our algorithm explicitly learns the value surface. We first consider the discrete-time problem and analyze neural network training error. We then turn to continuous problems and analyze the additional error due to the discretization of the underlying stochastic processes. Numerical experiments on high-dimensional American basket options and nonlinear utility maximization demonstrate that our method provides an efficient and scalable method for the multiple optimal stopping problem.
Paper Structure (13 sections, 12 theorems, 79 equations, 4 figures, 2 algorithms)

This paper contains 13 sections, 12 theorems, 79 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The general discrete-time setting
  3. The original problem
  4. The alternative problem
  5. Main results
  6. The original algorithm
  7. The alternative algorithm
  8. The convergence results
  9. Application to diffusion processes
  10. Discrete-time approximation
  11. Error for the alternative problem
  12. Numerical implementation
  13. Technical results

Key Result

Proposition 2.1

Assume the functions $c$ and $g$ are continuous. Then: ${\rm (i)}$ For all $({\boldsymbol{x}}, {\boldsymbol{i}}) \in \mathbb{R}^N \times \{0,1\}^N$, we have: where the inequality ${\boldsymbol{i}}' \le {\boldsymbol{i}}$ is understood coordinate-wise. ${\rm (ii)}$ There exists an optimal stopping strategy $\boldsymbol{\tau}^* \in \mathcal{T}_{p}^N$ for original-pb.

Figures (4)

  • Figure 1: Multi American Put with $N=5$: Comparison of the neural network value and the finite-difference value along the diagonal $x_1=x_2=x_3=x_4=x_5$ and on the plane $x_2=x_3=x_4=x_5$ at time $t=0$ (NN stands for the neural network, payoff for the value at terminal time, and FD for finite difference).
  • Figure 2: Multi American Put with $N=5$: error as a function of the time discretization (mean $\pm$ standard deviation).
  • Figure 3: Non-linear example with $N=5$: value functions obtained by neural network versus true value function along the diagonal $x_1 = \dots = x_N$.
  • Figure 4: Non-linear example with $N=5$: $L^2$ error as a function of the time discretization (mean $\pm$ standard deviation) and relative $L^2$ error.

Theorems & Definitions (14)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Lemma 4.2
  • Proposition 4.3
  • Remark 4.4
  • Proposition 4.5
  • ...and 4 more