Bellman Function Method in Analysis and Probability

The Bellman function method is an effective mathematical framework used to prove sharp inequalities across harmonic analysis, probability theory, and stochastic optimal control. This written course breaks down the underlying principles step by step, allowing you to understand how control theory connects to modern mathematical analysis. You will build a solid intuition for finding and verifying Bellman functions, starting from foundational definitions and progressing to classic applications in martingale theory and harmonic bounds. What you'll learn: - Understand the origins of the Bellman function method in stochastic optimal control - Learn the core terminology, concavity conditions, and boundary value setups - Apply Bellman techniques to derive sharp martingale inequalities and operator estimates - Master the step-by-step process of constructing Bellman candidate functions - Explore modern connections between dyadic models, harmonic analysis, and probabilistic methods - Practice verifying mathematical proofs using rigorous analytical reasoning The text guides you from core definitions and basic intuition through structured proofs and real-world analytical examples. This course is designed for learners with an interest in probability and mathematical analysis who want an accessible, text-based introduction with no advanced prior expertise in control theory required. Start reading today to deepen your mathematical toolkit.