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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.
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 ...
The concept of a stopped martingale leads to a series of important theorems, including, for example, the optional stopping theorem which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.
For convenience (see the proof below using the optional stopping theorem) and to specify the relation of the sequence (X n) n∈ and the filtration (F n) n∈ 0, the following additional assumption is often imposed:
The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of probability measure (see Girsanov's Theorem). If X is an R m valued semimartingale and f is a twice continuously differentiable function from R m to R n, then f(X) is a semimartingale. This is a consequence of Itō's ...
Retrieved from "https://en.wikipedia.org/w/index.php?title=Optional_stopping&oldid=240311808"This page was last edited on 22 September 2008, at 22:44
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 .
The earliest stopping time for reaching crossing point a, := {: =}, is an almost surely bounded stopping time. Then we can apply the strong Markov property to deduce that a relative path subsequent to τ a {\displaystyle \tau _{a}} , given by X t := W ( t + τ a ) − a {\displaystyle X_{t}:=W(t+\tau _{a})-a} , is also simple Brownian motion ...