Let be a martingale. Show that

So using the covariance formula, I know that

Now I know that I must use the property of martingale in which , but how should I mix that property in this problem? Should I take the double expectation on it? Thanks.

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- April 19th 2012, 08:22 AMtttcomraderProve that martingales have zero covariance
Let be a martingale. Show that

So using the covariance formula, I know that

Now I know that I must use the property of martingale in which , but how should I mix that property in this problem? Should I take the double expectation on it? Thanks. - April 29th 2012, 07:52 AMMooRe: Prove that martingales have zero covariance
Hello,

WLOG, we can suppose that i>j. Expand everything in

Then yes you have to take the double expectation for each term.

Just note that according to the martingale property for any , it would be useful to condition by the X with the lowest index.

For example, .

For , note that with the martingale property,

So that .