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Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

Author

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  • Nikolas Nüsken

    (Universität Potsdam)

  • Lorenz Richter

    (Freie Universität Berlin
    Brandenburgische Technische Universität Cottbus-Senftenberg)

Abstract

Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of iterative diffusion optimisation techniques, in particular considering applications in importance sampling and rare event simulation, and focusing on problems without diffusion control, with linearly controlled drift and running costs that depend quadratically on the control. More generally, our methods apply to nonlinear parabolic PDEs with a certain shift invariance. The choice of an appropriate loss function being a central element in the algorithmic design, we develop a principled framework based on divergences between path measures, encompassing various existing methods. Motivated by connections to forward-backward SDEs, we propose and study the novel log-variance divergence, showing favourable properties of corresponding Monte Carlo estimators. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.

Suggested Citation

  • Nikolas Nüsken & Lorenz Richter, 2021. "Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space," Partial Differential Equations and Applications, Springer, vol. 2(4), pages 1-48, August.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:4:d:10.1007_s42985-021-00102-x
    DOI: 10.1007/s42985-021-00102-x
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    References listed on IDEAS

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    1. David M. Blei & Alp Kucukelbir & Jon D. McAuliffe, 2017. "Variational Inference: A Review for Statisticians," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 859-877, April.
    2. Philipp Grohs & Fabian Hornung & Arnulf Jentzen & Philippe von Wurstemberger, 2018. "A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations," Papers 1809.02362, arXiv.org, revised Jan 2023.
    3. Baudoin, Fabrice, 0. "Conditioned stochastic differential equations: theory, examples and application to finance," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 109-145, July.
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    Cited by:

    1. Becker, Simon & Hartmann, Carsten & Redmann, Martin & Richter, Lorenz, 2022. "Error bounds for model reduction of feedback-controlled linear stochastic dynamics on Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 107-141.

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