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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.
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.
Where is Martingales used?
A martingale is a piece of equestrian tack designed to control a horse’s head carriage and act as an additional form of control besides, for example, the bit. 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.
What is a martingale trading strategy?
The Martingale system is a system of investing in which the dollar value of investments continually increases after losses, or the position size increases with the lowering portfolio size. The Martingale system was introduced by French mathematician Paul Pierre Levy in the 18th century.
How to calculate sum and product of martingale processes?
But what happens if one assumes that X is a martingale with respect to its own filtration F X and that Y is a martingale with respect to its own filtration F Y? (Recall that the filtration F Z of a process Z is defined by F n Z = σ ( { Z k; k ≤ n }) for every n .)
Is the sum of two independent martingales a martingale?
And yes, the sum of two independent martingales is a martingale but, here again, it might be wise to state the result with some care and, first of all, as mentioned by steveO in a comment, to specify the filtration (s) one is considering.
Is the expectation of a martingale constant in time?
Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time. More important, the expectation of a martingale is unaffected by optional sampling.
Which is a useful property of the martingales property?
The useful property of martingales is that we can verify the martingale property locally, by proving either that E [X t+1 |ℱ t] = X t or equivalently that E [X t+1 – X t |ℱ t] = E [X t+1 |ℱ t] – X t = 0. But this local property has strong consequences that apply across long intervals of time, as we will see below.