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In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating ...
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.
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 ...
Then is a stopping time with respect to the martingale (), so () is also a martingale, referred to as a stopped martingale. In particular, ( Y n ) n ∈ N {\displaystyle (Y_{n})_{n\in \mathbf {N} }} is a supermartingale which is bounded below, so by the martingale convergence theorem it converges pointwise almost surely to a random variable Y ...
If the sequence X = (X n) n∈ consists of symmetric random variables taking the values +1 and −1, then X is bounded, but the martingale M and the predictable process A are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale M unless the stopping time has ...
In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes.It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time.
Here Are My Tips for Stopping in 2025. Maria Leonard Olsen. December 17, 2024 at 11:57 AM ... then declines slowly over time. Think about the healthy life years you gain just from minimizing these ...
Down Panel: Paths of with an additional stopping criterion: the process is also stopped when it reaches a magnitude of =. This no longer suffers from gambler's ruin behavior, and is a martingale. Let W t be the Wiener process and T = min{ t : W t = −1 } the time of first hit of −1.