Bellman Equation

A Bellman equation, named after Richard E. Bellman, is a central tool of dynamic programming that breaks a complex optimization problem down into a sequence of smaller, connected stages. As a necessary condition for optimality, it expresses the value of a decision problem as the immediate payoff of an initial choice plus the value of whatever subproblems that choice leaves behind. This lets multi-period problems such as minimizing cost, maximizing profit or allocating a resource over time be solved recursively, by relating the value of a decision made now to the value of the decisions still to come. First developed for engineering control theory, the equation became a standard tool in economics, applied to problems from asset pricing to consumption and resource extraction, and in continuous time it becomes the Hamilton-Jacobi-Bellman equation, a partial differential equation used in optimal control.