## Is conditional expectation a martingale?

# 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.