Hamilton-Jacobi-Bellman equation
Quantitative Finance glossary · Quantitative Finance
Definition
Continuous-time dynamic-programming PDE for the value function V(t,x) of a stochastic control problem: 0 = (partial V)/(partial t) + sup_u!left mathcalL^u V + f(x,u) right, where mathcalL^u is the controlled infinitesimal generator and f the running reward. Underlies Merton's optimal consumption/investment problem (with closed-form CRRA solution), optimal execution (Almgren-Chriss), and reinforcement-learning-style hedging. Verification theorem links smooth solutions of HJB to optimal controls; viscosity solutions handle the non-smooth case.
Hamilton-Jacobi-Bellman equation in plain English
Imagine you're playing a long video game and at every instant you must choose a move to rack up the most points by the end — that's a 'stochastic control problem' (stochastic just means luck/randomness is involved). The Hamilton-Jacobi-Bellman (HJB) equation is the master rule that the 'best possible score from here on' obeys: it says at every moment you pick the action that gives you the best sum of the reward you grab right now plus how it improves your future prospects. In finance this cracks classic puzzles like how much of your money to spend versus invest today (Merton's problem), or how to sell a huge block of stock without crashing its price (optimal execution). The payoff is that instead of guessing move-by-move forever, you solve one equation for the whole strategy at once.
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