Analytical Concepts in Dynamic Programming
To understand the Bellman equation, several underlying concepts must be understood. First, any optimization problem has some objective--- minimizing travel time, minimizing cost, maximizing profits, maximizing utility, et cetera. The mathematical function that describes this objective is called the objective function.
Dynamic programming breaks a multi-period planning problem into simpler steps at different points in time. Therefore, it requires keeping track of how the decision situation is evolving over time. The information about the current situation which is needed to make a correct decision is called the state (See Bellman, 1957, Ch. III.2). For example, to decide how much to consume and spend at each point in time, people would need to know (among other things) their initial wealth. Therefore, wealth would be one of their state variables, but there would probably be others.
The variables chosen at any given point in time are often called the control variables. For example, given their current wealth, people might decide how much to consume now. Choosing the control variables now may be equivalent to choosing the next state; more generally, the next state is affected by other factors in addition to the current control. For example, in the simplest case, today's wealth (the state) and consumption (the control) might exactly determine tomorrow's wealth (the new state), though typically other factors will affect tomorrow's wealth too.
The dynamic programming approach describes the optimal plan by finding a rule that tells what the controls should be, given any possible value of the state. For example, if consumption (c) depends only on wealth (W), we would seek a rule that gives consumption as a function of wealth. Such a rule, determining the controls as a function of the states, is called a policy function (See Bellman, 1957, Ch. III.2).
Finally, by definition, the optimal decision rule is the one that achieves the best possible value of the objective. For example, if someone chooses consumption, given wealth, in order to maximize happiness (assuming happiness H can be represented by a mathematical function, such as a utility function), then each level of wealth will be associated with some highest possible level of happiness, . The best possible value of the objective, written as a function of the state, is called the value function.
Richard Bellman showed that a dynamic optimization problem in discrete time can be stated in a recursive, step-by-step form by writing down the relationship between the value function in one period and the value function in the next period. The relationship between these two value functions is called the Bellman equation.
Read more about this topic: Bellman Equation
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