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The Bellman equation is classified as a functional equation, because solving it means finding the unknown function , which is the value function. Recall that the value function describes the best possible value of the objective, as a function of the state x {\displaystyle x} .
For this simple system, the Hamilton–Jacobi–Bellman partial differential equation is (,) + {(,) (,) + (,)} =subject to the terminal condition (,) = (),As before, the unknown scalar function (,) in the above partial differential equation is the Bellman value function, which represents the cost incurred from starting in state at time and controlling the system optimally from then until time .
The iteration capability in Excel can be used to find solutions to the Colebrook equation to an accuracy of 15 significant figures. [3] [4] Some of the "successive approximation" schemes used in dynamic programming to solve Bellman's functional equation are based on fixed-point iterations in the space of the return function. [5] [6]
Bellman's contribution is remembered in the name of the Bellman equation, a central result of dynamic programming which restates an optimization problem in recursive form. Bellman explains the reasoning behind the term dynamic programming in his autobiography, Eye of the Hurricane: An Autobiography: I spent the Fall quarter (of 1950) at RAND ...
Backward induction was first utilized in 1875 by Arthur Cayley, who discovered the method while attempting to solve the secretary problem. [2] In dynamic programming, a method of mathematical optimization, backward induction is used for solving the Bellman equation.
Originally introduced by Richard E. Bellman in (Bellman 1957), stochastic dynamic programming is a technique for modelling and solving problems of decision making under uncertainty. Closely related to stochastic programming and dynamic programming, stochastic dynamic programming represents the problem under scrutiny in the form of a Bellman ...
Some classes of functional equations can be solved by computer-assisted techniques. [vague] [4] In dynamic programming a variety of successive approximation methods [5] [6] are used to solve Bellman's functional equation, including methods based on fixed point iterations.
There are generally two approaches to solving optimal stopping problems. [4] When the underlying process (or the gain process) is described by its unconditional finite-dimensional distributions , the appropriate solution technique is the martingale approach, so called because it uses martingale theory, the most important concept being the Snell ...