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Xianhua Peng, Wu Guo · 2026-07-05
The paper introduces a new deep-learning method (Certainty Equivalent Learning, or CEL) for solving high-dimensional dynamic programming problems where preferences follow recursive utility (like Epstein-Zin), which are hard to solve because of a difficult-to-evaluate certainty-equivalent term. The method uses neural networks to jointly approximate value, policy, and certainty-equivalent functions without needing meshes, Euler equations, or differentiability. Tested on several economic problems, it produced accurate solutions with small out-of-sample errors compared to closed-form and value-function-iteration benchmarks.
Why it matters: For quants working on multi-asset allocation or macro-finance models, this offers a computational tool to solve dynamic portfolio and control problems with realistic recursive preferences in high dimensions where traditional methods break down. One application demonstrated is multivariate strategic asset allocation, which could matter for long-horizon portfolio construction under sophisticated risk preferences.
⚠ This is a computational methods paper validated against theoretical/numerical benchmarks, not a trading strategy or live-market result, and requires substantial modeling and machine-learning expertise to apply.
We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recursive utility does not have an explicit representation and the Bellman equation contains a certainty equivalent that is difficult to evaluate. The CEL algorithm learns this certainty-equivalent value directly with neural networks and jointly approximates value functions, policy functions, and certainty-equivalent functions. The CEL algorithm is mesh-free and simulation-based, allowing high-dimensional state and control spaces, and does not rely on Euler equations, first-order conditions, or differentiability of the state transition function. The CEL algorithm also works for dynamic programming problems with expected utility as expected utility is a special case of recursive utility. We apply the CEL to discounted linear exponential quadratic Gaussian control, small-noise robust control, Epstein-Zin DSGE, and multivariate strategic asset allocation problems. Compared with closed-form and VFI-based benchmarks, the CEL delivers accurate value and policy approximations, remains effective in high-dimensional problems, achieves accuracy comparable to VFI in the small-noise robust-control case, and produces out-of-sample Bellman errors and Euler or first-order residuals that are in the range from 1.0e-4 to 1.0e-3 for most problems.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.