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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 ...
In some contexts the concept of stopping time is defined by requiring only that the occurrence or non-occurrence of the event τ = t is probabilistically independent of X t + 1, X t + 2, ... but not that it is completely determined by the history of the process up to time t. That is a weaker condition than the one appearing in the paragraph ...
A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process. In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener.
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 ...
Craigmile P.F. (2003), "Simulating a class of stationary Gaussian processes using the Davies–Harte Algorithm, with application to long memory processes", Journal of Times Series Analysis, 24: 505–511. Dieker, T. (2004). Simulation of fractional Brownian motion (PDF) (M.Sc. thesis)
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.
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 ...