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For example, suppose the last example had instead used a 'one-sided' stopping time, so that stopping only occurred at +m, not at −m. The value of X at this stopping time would therefore be m. Therefore, the expectation value E[X τ] must also be m, seemingly in violation of the theorem which would give E[X τ] = 0. The failure of the optional ...
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:
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.
The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem. [ 3 ] However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip ...
The proof can also be phrased in the language of stochastic processes so as to become a corollary of the powerful theorem that a stopped submartingale is itself a submartingale. [2] In this setup, the minimal index i appearing in the above proof is interpreted as a stopping time .
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 .
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