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2. Martingale (probability theory) - Wikipedia

   en.wikipedia.org/wiki/Martingale_(probability...

   A convex function of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that X n 2 − n is a martingale). Similarly, a concave function of a martingale is a supermartingale.

3. Risk-neutral measure - Wikipedia

   en.wikipedia.org/wiki/Risk-neutral_measure

   In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.

4. Martingale (tack) - Wikipedia

   en.wikipedia.org/wiki/Martingale_(tack)

   A martingale is any of several designs of tack that are used on horses to control head carriage. Martingales may be seen in a wide variety of equestrian disciplines, both riding and driving. Rules for their use vary widely; in some disciplines they are never used, others allow them for schooling but not in judged performance, and some ...

5. Azuma's inequality - Wikipedia

   en.wikipedia.org/wiki/Azuma's_inequality

   In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences. Suppose {: =,,,, …} is a martingale (or super-martingale) and

6. Optional stopping theorem - Wikipedia

   en.wikipedia.org/wiki/Optional_stopping_theorem

   Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true.

7. Martingale (betting system) - Wikipedia

   en.wikipedia.org/wiki/Martingale_(betting_system)

   A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails.

8. Doob's martingale inequality - Wikipedia

   en.wikipedia.org/wiki/Doob's_martingale_inequality

   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.

9. Doob's martingale convergence theorems - Wikipedia

   en.wikipedia.org/wiki/Doob's_martingale...

   The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge. As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening.