Common questions

What is the expectation of a martingale?

What is the expectation of a martingale?

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.

What is linearity of expectation?

Linearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent. The expected value of a random variable is essentially a weighted average of possible outcomes.

What is the expectation of variance?

Given a random variable, we often compute the expectation and variance, two important summary statistics. The expectation describes the average value and the variance describes the spread (amount of variability) around the expectation.

How do you calculate conditional expectations?

The conditional expectation, E(X |Y = y), is a number depending on y. If Y has an influence on the value of X, then Y will have an influence on the average value of X. So, for example, we would expect E(X |Y = 2) to be different from E(X |Y = 3).

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.

How can you tell if its a martingale?

In general, if Yt+1-Yt = bt(Xt+1-Xt) where (Xt,ℱt) is a martingale and bt is measurable ℱt, then Yt is also a martingale with respect ℱt.

Do expectations multiply?

Multiplying a random variable by any constant simply multiplies the expectation by the same constant, and adding a constant just shifts the expectation: E[kX+c] = k∙E[X]+c . For any event A, the conditional expectation of X given A is defined as E[X|A] = Σx x ∙ Pr(X=x | A) .

What are the properties of expectation?

The following properties of expectation apply to discrete, continuous, and mixed random variables:

  • Indicator function. The expectation of the indicator function is a probability: (5.56)
  • Linearity. Expectation is a linear operator: (5.58)
  • Nonnegative.
  • Symmetry.
  • Independence.

What is expectation function?

Using expectation, we can define the moments and other special functions of a random variable. Definition 2 Let X and Y be random variables with their expectations µX = E(X) and µY = E(Y ), and k be a positive integer. 1. The kth moment of X is defined as E(Xk). If k = 1, it equals the expectation.

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.

Why are martingales called martingales?

The word martingale came from a group of betting strategies that were popular in France in the 18th century. In a simple game where a gambler wins if a coin comes up heads and loses if it comes up tails (pt = ph = 1/2, assuming a fair coin) the martingale strategy had him double his bet every time he lost.

How does the law of iterated expectation work?

The Law of Iterated Expectation states that the expected value of a random variable is equal to the sum of the expected values of that random variable conditioned on a second random variable.

When does the expectation of x equal the expected value?

In other words, if X and Y are random variables that take different values with probability zero, then the expectation of X will equal the expectation of Y. . In particular, for a random variable . A well defined expectation implies that there is one number, or rather, one constant that defines the expected value.

Which is the decomposition of variance in iterated expectations?

Decomposition of variance (Wooldridge, p. 31) • Proof that var(y) = var. x[E(y|x)]+E. x[var(y|x)] (i.e., the variance of y decomposes into the variance of the conditional mean plus the expected variance around the conditional mean).

Which is the best description of the law of total expectation?

The proposition in probability theory known as the law of total expectation, the law of iterated expectations ( LIE ), the tower rule, Adam’s law, and the smoothing theorem, among other names, states that if is a random variable whose expected value is defined, and is any random variable on…