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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 ...
A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time. Stopping times occur in decision theory, and the optional stopping theorem is an important result ...
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:
In mathematics, the theory of optimal stopping [1] [2] or early stopping [3] is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost.
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. [1]
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Optical theorem ; Optional stopping theorem (probability theory) Orbit theorem (Nagano–Sussmann) (control theory) Orbit-stabilizer theorem (group theory) Ore's theorem (graph theory) Orlicz–Pettis theorem (functional analysis) Ornstein theorem (ergodic theory) Oseledec theorem (ergodic theory) Osterwalder–Schrader theorem