--- title: Hamilton-Jacobi-Bellman Equation: Reinforcement Learning... description: A deep dive into the Hamilton-Jacobi-Bellman (HJB) equation, tracing its roots from Bellman's 1952 dynamic programming work through continuous-time... canonical: https://daily.dev/posts/hamilton-jacobi-bellman-equation-reinforcement-learning-and-diffusion-models-um6oohsnn twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Hamilton-Jacobi-Bellman Equation: Reinforcement Learning and Diffusion Models | daily.dev og:description: A deep dive into the Hamilton-Jacobi-Bellman (HJB) equation, tracing its roots from Bellman's 1952 dynamic programming work through continuous-time... og:url: https://daily.dev/posts/hamilton-jacobi-bellman-equation-reinforcement-learning-and-diffusion-models-um6oohsnn og:image: https://api.daily.dev/og/posts/Um6oohsNn.png og:image:alt: Hamilton-Jacobi-Bellman Equation: Reinforcement Learning and Diffusion Models og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Hamilton-Jacobi-Bellman Equation: Reinforcement Learning and Diffusion Models **[Hacker News](https://daily.dev/sources/hn)** · 9 min read · 0 upvotes · 0 comments ## Summary A deep dive into the Hamilton-Jacobi-Bellman (HJB) equation, tracing its roots from Bellman's 1952 dynamic programming work through continuous-time reinforcement learning to modern diffusion models. Covers the derivation of HJB for both deterministic and stochastic controlled diffusions, introduces the continuous-time Q-function, and presents a neural policy iteration algorithm using MLPs for value and policy networks. Benchmarks include the linear-quadratic regulator and Merton's portfolio problem. The post also shows how diffusion model training can be recast as a finite-horizon stochastic optimal control problem, with the score function emerging as the optimal control law. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Similar posts on daily.dev - [Bellman Equations and Dynamic Programming in RL](https://daily.dev/posts/bellman-equations-and-dynamic-programming-in-rl-vuacrm8zo) · Daily Dose of Data Science \| Avi Chawla \| Substack · 3 upvotes · 0 comments - [Bellman Equation: A Comprehensive Guide](https://daily.dev/posts/bellman-equation-a-comprehensive-guide-nfsuexrh9) · Medium · 0 upvotes · 0 comments - [A Tale of Dynamic Programming \(2022\)](https://daily.dev/posts/a-tale-of-dynamic-programming-2022--demkthpmp) · Hacker News · 0 upvotes · 0 comments - [The Reinforcement Learning Handbook: A Guide to Foundational Questions](https://daily.dev/posts/the-reinforcement-learning-handbook-a-guide-to-foundational-questions-0vv0xujq8) · Towards Data Science · 1 upvotes · 0 comments - [Overcoming Dimensional Factorization Limits in Discrete Diffusion Models through Quantum Joint Distribution Learning](https://daily.dev/posts/overcoming-dimensional-factorization-limits-in-discrete-diffusion-models-through-quantum-joint-distr-xff3xbyei) · Nature · 0 upvotes · 0 comments --- Tags: [#neural-networks](https://daily.dev/tags/neural-networks), [#reinforcement-learning](https://daily.dev/tags/reinforcement-learning), [#diffusion-models](https://daily.dev/tags/diffusion-models) [View this post on daily.dev](https://daily.dev/posts/hamilton-jacobi-bellman-equation-reinforcement-learning-and-diffusion-models-um6oohsnn) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Hamilton-Jacobi-Bellman Equation: Reinforcement Learning and Diffusion Models","url":"https://daily.dev/posts/hamilton-jacobi-bellman-equation-reinforcement-learning-and-diffusion-models-um6oohsnn","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/hamilton-jacobi-bellman-equation-reinforcement-learning-and-diffusion-models-um6oohsnn"},"datePublished":"2026-03-30T14:02:06.893Z","dateModified":"2026-03-30T14:02:43.692Z","description":"A deep dive into the Hamilton-Jacobi-Bellman (HJB) equation, tracing its roots from Bellman's 1952 dynamic programming work through continuous-time...","image":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/07a3416277167f84584a345d044e801c?_a=AQAEuop","thumbnailUrl":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/07a3416277167f84584a345d044e801c?_a=AQAEuop","isAccessibleForFree":true,"articleSection":"Hacker News","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Hacker News","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/hn","url":"https://daily.dev/sources/hn"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/hamilton-jacobi-bellman-equation-reinforcement-learning-and-diffusion-models-um6oohsnn","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"neural-networks,reinforcement-learning,diffusion-models","timeRequired":"PT9M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Hacker News","item":"https://daily.dev/sources/hn"},{"@type":"ListItem","position":3,"name":"Hamilton-Jacobi-Bellman Equation: Reinforcement Learning and Diffusion Models"}]} ```