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. …

2024年1月9日 · In the gambling world such a system is called a martingale, which explains the origin of the mathematical term "martingale" . One of the basic facts of the theory of martingales is that the structure of a martingale (submartingale) $ X = ( X _ {t} , {\mathcal F} _ {t} ) $ is preserved under a random change of time.

2022年4月23日 · Indeed, martingales are of fundamental importance in modern probability theory. Here are two related definitions, with equality in the martingale condition replaced by inequalities. Suppose again that the process X and the filtration F satisfy the basic assumptions above.

In words, prices follow a martingale, and successive price changes are mutually uncorrelated. This implies that if “prices are properly anticipated,” all the information in the past price series that is useful for forecasting next period’s expected price is contained in the current price.

2018年3月21日 · What is a Martingale? A martingale is model of a fair game. It is a sequence of random variables x 0, x 1, x 2 …x n with one important property: the conditional expectation of x n+1 given x 0, x 1, x 2 …x n is always just x n. In other words, it is a sequence of random variables such that for any time n: E (X n + 1 | X 1 …, X n) = X n.

Informally a martingale is simply a stochastic process Mt defined on some probability space (Ω, F, P) and indexed by some ordered set T that is “con-ditionally constant,” i.e., whose predicted value at any future time s > t is the same as its present value at the time t of prediction.

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.

To motivate this definition, imagine that Xn represents the price of a stock on day n. In this context, the martingale condition states informally that “The expected value of the stock tomorrow, given all I know today, is the value of the stock today.”

Definition of Conditional Expectation. For random variables defined on discrete proba-bility spaces, conditional expectation can be defined in an elementary manner: In particular, the conditional expectation of a discrete random variable X given the value y of another dis-c. the set of all possible values x of X . Note that.

2025年1月31日 · The term was first used to describe a type of wagering in which the bet is doubled or halved after a loss or win, respectively. The concept of martingales is due to Lévy, and it was developed extensively by Doob. A one-dimensional random walk with steps equally likely in either direction () is an example of a martingale.