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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
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.
Although MT5 was introduced in 2009, according to a study conducted in September 2019, MetaTrader 4 was still the most popular Forex trading platform in the world at the time. [ 10 ] On September 24, 2022, it was reported that MetaTrader 4 and MetaTrader 5 platforms had been removed from Apple's App Store but was still available on the Android ...
A discrete-time version of the theorem is given below, with 0 denoting the set of natural integers, including zero.. Let X = (X t) t∈ 0 be a discrete-time martingale and τ a stopping time with values in 0 ∪ {∞}, both with respect to a filtration (F t) t∈ 0.
where is the Radon–Nikodym derivative of with respect to , and therefore is still a martingale. [3] If in a financial market there is just one risk-neutral measure, then there is a unique arbitrage-free price for each asset in the market.
The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that ( M t ) 0 ≤ t < ∞ {\displaystyle \left(M_{t}\right)_{0\leq t<\infty }} is a Q-martingale process, whose volatility σ t {\displaystyle \sigma _{t}} is always non-zero.
By construction, this implies that if is a martingale, then = will be an MDS—hence the name. The MDS is an extremely useful construct in modern probability theory because it implies much milder restrictions on the memory of the sequence than independence , yet most limit theorems that hold for an independent sequence will also hold for an MDS.
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, [1] also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random ...