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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.
Pandya theorem (nuclear physics) Pomeranchuk's theorem ; Reeh–Schlieder theorem (local quantum field theory) Spin–statistics theorem ; Stone–von Neumann theorem (functional analysis, representation theory of the Heisenberg group, quantum mechanics) Supersymmetry nonrenormalization theorems ; Vafa–Witten theorem
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 mathematics, a point process is a random element whose values are "point patterns" on a set S.While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points.
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 ...