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Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem .
Graphs of probabilities of getting the best candidate (red circles) from n applications, and k/n (blue crosses) where k is the sample size. The secretary problem demonstrates a scenario involving optimal stopping theory [1] [2] that is studied extensively in the fields of applied probability, statistics, and decision theory.
What stopping rule minimizes the expected rank of the selected observation, and what is its corresponding value? The general solution to this full-information expected rank problem is unknown. The major difficulty is that the problem is fully history-dependent, that is, the optimal rule depends at every stage on all preceding values, and not ...
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. [1] 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 ...
Suppose further that the walk stops if it reaches 0 or m ≥ a; the time at which this first occurs is a stopping time. If it is known that the expected time at which the walk ends is finite (say, from Markov chain theory), the optional stopping theorem predicts that the expected stop position is equal to the initial position a.
The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. [4] [5] [6] The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. [7] In discrete-time problems, the analogous difference equation is usually referred to as the ...
The "index policy" induced by the Gittins index, consisting of choosing at any time the stochastic process with the currently highest Gittins index, is the solution of some stopping problems such as the one of dynamic allocation, where a decision-maker has to maximize the total reward by distributing a limited amount of effort to a number of ...
In dynamic programming, a method of mathematical optimization, backward induction is used for solving the Bellman equation. [3] [4] In the related fields of automated planning and scheduling and automated theorem proving, the method is called backward search or backward chaining. In chess, it is called retrograde analysis.