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A quantitative averaged convergence estimate is proved for the gap between the discrete-time value and the value induced by an approximately optimized neural policy. The bound cleanly disentangles policy approximation near optimality, compact-set localization of stochastic trajectories, and training optimization tolerance, without requiring transition-density assumptions. Degenerate diffusions and deterministic controlled dynamics are handled in one unified framework, supported by numerical experiments.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/neural-feedback-approximation-for-stochastic-control-with-degenerate-diffusions-error-estimates-and-numerical-analysis/85865/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/neural-feedback-approximation-for-stochastic-control-with-degenerate-diffusions-error-estimates-and-numerical-analysis/85865.png","ImageObject",300,407,{"name":92,"@type":93},"Ezra","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-20","2026-07-16",true,{"@type":102,"interactionType":103,"userInteractionCount":56},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does the paper address?","Question",{"text":112,"@type":113},"The paper studies finite-horizon stochastic optimal control and approximates the resulting time-discrete problem using neural-network feedback policies.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is convergence error estimated?",{"text":117,"@type":113},"It provides a quantitative convergence estimate in an averaged sense for the error between the time-discrete value and the value generated by an approximately optimized neural policy.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the error bound separate?",{"text":121,"@type":113},"The bound separates near-optimal feedback policy approximation, localization of stochastic trajectories on compact sets, and optimization tolerance arising from training.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},85865,1784206784,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":56,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},2336474466412,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","arXiv :2607 . 10254v1 [math .OC] 11 Jul 2026  \nNeural feedback approximation for stochastic control with degenerate diffusions: error estimates and numerical analysis  \nOlivier Bokanowski∗ Jean-François Chassagneux† Marco Scaratti‡§  \nXavier Warin¶  \nJuly 14, 2026  \nAbstract  \nWe study finite-horizon stochastic optimal control problems and approximate the resulting time-discrete formulation by a direct policy-learning problem over neural-network feedback maps. We prove a quantitative convergence estimate, in an averaged sense, for the error between the time-discrete value and the value induced by an approximately optimized neural policy. The bound separates the approximation of near-optimal feedback policies, the localization of stochastic trajectories on compact sets, and the optimization tolerance in training. The analysis does not require transition-density assumptions and covers possibly degenerate diffusions and deterministic controlled dynamics in a unified framework. Numerical experiments are provided for a degenerate stochastic radial target problem, a Hamilton–Jacobi–Bellman benchmark, and a gas storage problem, illustrating the approach and separating the main error sources: time discretization, restriction to piecewise-constant policies, neural-network approximation, and Monte Carlo evaluation.  \nKeywords. Stochastic optimal control; degenerate diffusions; neural networks; feedback control approximation; error estimates; time-discrete approximation.  \nMSC 2020 . Primary: 93E20, 49M25, 65C30 . Secondary: 68T07, 60H10 .  \n1 Introduction  \nStochastic optimal control problems arise in many applications in which decisions must be made dynamically under uncertainty, including finance, economics, energy management, operations research, engineering, and models of large interacting populations. In continuous time, these problems are usually formulated through controlled stochastic differential equations and have been studied through several classical approaches: among others, the dynamic programming principle (DPP) and the associated Hamilton–Jacobi–Bellman (HJB) equation [17 , 36], weak formulations based on martingale methods, changes of measure, backward stochastic differential equations (BSDEs) [16], and stochastic maximum principle methods based on Hamiltonians and adjoint equations [35 , 52] . Despite this well-established theory, the numerical solution of stochastic control problems remains challenging in moderate and high dimensions. Grid-based  \n∗ Université Paris Cité, Laboratoire Jacques-Louis Lions, [olivier.bokanowski@u-paris.fr. This](olivier.bokanowski@u-paris.fr. This) research benefited from the support of the FMJH Program PGMO and from the support to this program from EDF.  \n†ENSAE, CREST and Institut Polytechnique de Paris, jean-francois.chassagneux@ensae.fr. This research has benefited from the support of the ANR Project ReLISCoP (ANR-21-CE40-0001) .  \n‡Corresponding [author: marco.scaratti@univr.it](author: marco.scaratti@univr.it).  \n§ University of Verona and Université Paris Cité, Laboratoire Jacques-Louis Lions.  \n¶ EDF Lab Paris-Saclay and FiME, Laboratoire de Finance des Marchés de l’Energie, 91120 Palaiseau, France, [xavier.warin@edf.fr](xavier.warin@edf.fr).  \napproximations of HJB equations suffer from the curse of dimensionality, while computing accurate feedback controls is particularly difficult when the dynamics are degenerate or when the diffusion coefficients are controlled.  \nIn recent years, neural-network methods have provided a flexible computational tool for high-dimensional stochastic control and related nonlinear PDEs. Broadly speaking, the existing literature can be divided into two main families. The first one is value-based: the neural network is used to approximate the value function, its gradient, a BSDE component, or the residual of a PDE. The second one is policy-based: the feedback control itself is parameterized by a neural network and optimized directly through simulated trajectories. These","cbCaigbJ8WwjHoZ3","https://ap.wps.com/l/cbCaigbJ8WwjHoZ3","pdf",795732,36,"English","# Abstract\n# Introduction","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper studies finite-horizon stochastic optimal control and approximates the resulting time-discrete problem using neural-network feedback policies.\"},{\"question\":\"How is convergence error estimated?\",\"answer\":\"It provides a quantitative convergence estimate in an averaged sense for the error between the time-discrete value and the value generated by an approximately optimized neural policy.\"},{\"question\":\"What does the error bound separate?\",\"answer\":\"The bound separates near-optimal feedback policy approximation, localization of stochastic trajectories on compact sets, and optimization tolerance arising from training.\"}]","Neural feedback approximation for stochastic control with degenerate diffusions: error estimates and numerical analysis | PDF",91]