A Bellman equation (also known as a dynamic programming equation), named after its discoverer, Richard Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the value of a decision problem at a certain point in time in terms of the payoff from some initial choices and the value of the remaining decision problem that results from those initial choices. This breaks a dynamic optimization problem into simpler subproblems, as Bellman's Principle of Optimality prescribes.
The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory.
Almost any problem which can be solved using optimal control theory can also be solved by analyzing the appropriate Bellman equation. However, the term 'Bellman equation' usually refers to the dynamic programming equation associated with discrete-time optimization problems. In continuous-time optimization problems, the analogous equation is a partial differential equation which is usually called the Hamilton-Jacobi-Bellman equation.
Read more about Bellman Equation: Analytical Concepts in Dynamic Programming, Solution Methods, Applications in Economics, Example
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