Is conditional expectation a martingale?

Is conditional expectation a martingale?

Some Examples of Martingales. Let X be any integrable random variable. Then the sequence Xn defined by Xn = E(X|Fn) is a martingale, by the Tower Property of conditional expectation. Yj is a martingale, as we have seen.

What are the findings of martingale?

If p is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler’s fortune over time is a martingale. If p is less than 1/2, the gambler loses money on average, and the gambler’s fortune over time is a supermartingale.

Are martingales independent?

Martingales as sums of uncorrelated random variables. , where E[Δi|Δ1…Δi-1] = E[Xi-Xi-1|ℱi] = 0. In other words, Δi is uncorrelated with Δ1…Δi-1. This is not quite as good a condition as independence, but is good enough that martingales often act very much like sums of independent random variables.

What is martingale hypothesis?

The martingale hypothesis defines that the level of any variable in is equal to the price of the same variable in t using all the past information set. Hence, the expectation on a future variation of price influenced by the price history set must be equal to zero.

What is a zero mean martingale?

We show that M(·) is a zero mean martingale. Because it is constructed from a counting process, it is referred to as a counting process martingale. = X(t) ∀s, t ≥ 0. X(·) is a sub-martingale if above holds but with “=” in (b) replaced by “≥”; called a super-martingale if “=” replaced by “≤”.

Is the product of two martingales a martingale?

The product of two independent martingales is a martingale–or rather it is or it is not, depending on the precise formulation of the hypothesis! When it is, one says that the martingales are orthogonal.

Why it is called martingale?

Doob is the one who really made the name popular (in addition to proving many fundamental results). He got the name from a thesis by Ville. A martingale is the name for a Y-shaped strap used in a harness — it runs along the horse’s chest and then splits up the middle to join the saddle.

Is a random walk martingale?

A Martingale process is similar to a one-dimensional random walk.

Is W 3 a martingale?

However the first piece on the LHS in not a martingale and thus W3(t) is not a martingale.

Why is martingale important?

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.

Are two Brownian motions independent?

some filtration, and since the quadratic variations are [X,X]t=t, [Y,Y]t=t and [X,Y]t=0, Lévy’s characterisation of Brownian motion gives that (X,Y) is a two dimensional Brownian motion, in particular X,Y are independent.

Is the Martingale system profitable?

Conclusion. The Martingale system eventually leads to large losses that wipe out all of your short-term profits. But if you know how it works and the long-term dangers, you can still use this system for fun.

What is the property of being a martingale?

It is important to note that the property of being a martingale involves both the filtration and the probability measure (with respect to which the expectations are taken).

Which is an example of a martingale in probability theory?

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.

Which is a martingale under the unified neutral theory?

This sequence is a martingale under the unified neutral theory of biodiversity and biogeography. If { Nt : t ≥ 0 } is a Poisson process with intensity λ, then the compensated Poisson process { Nt − λ t : t ≥ 0 } is a continuous-time martingale with right-continuous/left-limit sample paths.

Which is a convex function of a martingale?

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.