## Two Period Economy And Arbitrage - Computer Science.

The Martingale property states that the future expectation of a stochastic process is equal to the current value, given all known information about the prior events. Both of these properties are extremely important in modeling asset price movements.

Martingale process. Any martingale process is a sequence of random variables that satisfies.(1) The discounted stock price under the risk neutral probability measures is a martingale process. The risk neutral probabilities are chosen to enforce the fact. i.e.(2) here the indicates the expected value under risk neutral measure.

N.B.: A martingale is the mathematical description of a fair game: expected net gain is zero and independent of history. This is modeled as a random walk without drift: the expected position of a martingale sometime in the future is exactly where it is now. A local martingale is a stochastic process that is locally a martingale.

Robust pricing and hedging under trading restrictions and the emergence of local martingale models. expected value of the payo under a risk-neutral measure is both classical and well understood. In a. of supermartingale measures is simply the class of martingale measures.

If Q is a martingale measure and H is a self nancing trading strategy, then V, the discounted value process corresponding to H, is a martingale under Q. Completeness theorem If the market is complete, that is, all contingent claims can be replicated, then. their discounted expected values under the risk neutral (equivalent martingale).

Therefore, we can consider a martingale as a process whose expected value, conditional on some potential information, is equal to the value revealed by the last available information. Similarly, a submartingale represents a favorable game because the expected fortune increases in the future, while a supermartingale represents an unfavorable game because the expected fortune decreases in the.

A stopped Brownian motion as an example for a martingale. In probability theory, a martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time t, given all the observations up to some earlier time s, is equal to the observation at that earlier time s.A martingale is a model of a fair game (disambiguation.