arXiv 5 Jul 2026 · Finance — Computational
arXiv:2607.04278 · PDF · DOI · OpenAlex · Extracted main text
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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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Hansen \ Sargent (1995) Discounted linear exponential quadratic gaussian control, IEEE Transactions on Automatic control 40(5): 968–971 | 0.928 | 4 | 3 | 100% |
| 2 | Epstein \ Zin (1989) Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework, Econometrica 5… | 0.843 | 3 | 3 | 100% |
| 3 | Hansen \ Sargent (2013) Recursive Models of Dynamic Linear Economies, Princeton University Press | 0.737 | 3 | 2 | 100% |
| 4 | Judd (1998) Numerical Methods in Economics, MIT Press, Cambridge, MA | 0.737 | 3 | 2 | 100% |
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| 6 | Duffie \ Epstein (1992) Stochastic differential utility, Econometrica 60: 353–394 | 0.644 | 2 | 2 | 100% |
| 7 | Weil (1989) The equity premium puzzle and the risk-free rate puzzle, Journal of Monetary Economics 24(3): 401–421 | 0.644 | 2 | 2 | 100% |
| 8 | Weil (1990) Nonexpected utility in macroeconomics, The Quarterly Journal of Economics 105(1): 29–42 | 0.644 | 2 | 2 | 100% |
| 9 | Campbell, Chan \ Viceira (2003) A multivariate model of strategic asset allocation, Journal of financial economics 67(1): 41–80 | 0.644 | 2 | 2 | 100% |
| 10 | Hansen \ Sargent (2008) Robustness, Princeton university press | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 80 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Deep Learning for Dynamic Programming with Recursive Utility Using First-order Conditions | 0.511 | 2 | 1 |