Martingals

Martingales are a class of stochastic processes that are widely used in probability theory and mathematical finance. They are named after the French mathematician Jean-Charles de la Vallée-Poussin, who introduced the concept in the early 20th century. A martingale is a sequence of random variables where the expected value of the next term, given all previous terms, is equal to the current term. This property is known as the martingale property.

Formally, a sequence of random variables {X_n} is a martingale with respect to a filtration {F_n} if, for all n, E[X_{n+1} | F_n] = X_n, where E denotes the expected value and F_n is the sigma-algebra generated by the first n random variables. The filtration {F_n} represents the information available up to time n.

Martingales have several important properties. They are closed under taking limits, meaning that the limit of a sequence of martingales is also a martingale. They also have a stopping time property, which states that if a stopping time T is used to stop a martingale, then the stopped process is still a martingale.

Martingales are used in various fields, including finance, where they are used to model the evolution of stock prices. They are also used in gambling theory, where they are used to analyze the behavior of betting strategies. In probability theory, martingales are used to study the behavior of random walks and other stochastic processes.