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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.
In decision theory, the odds algorithm (or Bruss algorithm) is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the odds strategy, and the importance of the odds strategy lies in its optimality, as explained below.
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.
However, the Bellman Equation is often the most convenient method of solving stochastic optimal control problems. For a specific example from economics, consider an infinitely-lived consumer with initial wealth endowment a 0 {\displaystyle {\color {Red}a_{0}}} at period 0 {\displaystyle 0} .
A simple suboptimal rule, which performs almost as well as the optimal rule within the class of memoryless stopping rules, was proposed by Krieger & Samuel-Cahn. [7] The rule stops with the smallest i {\displaystyle i} such that R i < i c / ( n + i ) {\displaystyle R_{i}<ic/(n+i)} for a given constant c, where R i {\displaystyle R_{i}} is 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 ...
Example of a stopping time: a hitting time of Brownian motion.The process starts at 0 and is stopped as soon as it hits 1. In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time [1]) is a specific type of “random time”: a random variable whose value is interpreted as the time at ...