Why are martingales so important?

Why are martingales so important?

Essentially, the martingale property ensures that in a “fair game”, knowledge of the past will be of no use in predicting future winnings. These properties will be of fundamental importance in regard to defining Brownian motion, which will later be used as a model for an asset price path.

What are martingales?

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.

How do you prove a local martingale is a martingale?

If (Xt)t≥0 ( X t ) t ≥ 0 is a local martingale which is dominated by an integrable random variable, in the sense that Esupt≥0|Xt|<∞ E sup t ≥ 0 | X t | < ∞ , then (Xt)t≥0 ( X t ) t ≥ 0 is a martingale, and in fact a uniformly integrable martingale.

Is constant a martingale?

The constant, deterministic sequence Xn = 7 is a martingale: in this case E[Xn+1|Fn]=7= Xn for all n ≥ 0.

Can you martingale the stock market?

Martingale in stock market trading In stock markets, the Martingale strategy is implemented when a trader keeps doubling his position size till he makes a winning trade. There are variations on this, where the trader increases his position each time he loses but not necessarily by doubling it.

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.

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.

What is a positive martingale?

The sequence is called a positive martingale if the inequality is replaced by an equality. This definition makes sense as the conditional expectation of an arbitrary positive real-valued r.v. was defined in the first chapter (Lemma 1-2-9).

Is a local martingale integrable?

There do exist integrals, with respect to local martingales, which are not locally integrable and, hence, cannot themselves be local martingales.

Is W 3 a martingale?

The second piece on the LHS is an Ito integral and thus a martingale. However the first piece on the LHS in not a martingale and thus W3(t) is not a martingale.

What is a martingale and why is it important?

Martingales are truly fundamental objects. Here are some of my favorite facts about them: 1. A martingale is the probabilistic extension of a flat line. In other words, a flat line is the martingale when the probability space is trivial. 2. Martingales are critical in models of gambling (and by extension, stochastic control and optimal stopping).

What are the different types of martingales for horses?

It prevents a horse from throwing its head so high that the rider gets hit in the face by the horse’s poll or upper neck. There are three main types of martingales: the standing, the running, and the German martingale.

What is the formula for a martingale process?

Intuitively, a martingale is a stochastic process for which the conditional expectation of its future value, given the information accumulated up to now, equals to its current value. E ( X t | F s) = X s for all t ≥ s ≥ 0.

Which is better a standing or running martingale?

A Running Martingale is considered to be safer than its counterpart the Standing Martingale, as the rider has more control over allowing or restricting the horse’s range of head movement. When using a Running Martingale, it’s important to use rein stops.