Difference sequence non-negative expectations

Reference no: EM131830319

Prove the theorem.

Let {U_n} be {F_n}-adapted and integrable, and set

2321_adapted and integrable.png

(i) {(X_n, F_n), n ≥ 0} is a martingale iff {(U_n, F_n), n ≥ 0} is a martingale difference sequence, a submartingale iff {(U_n, F_n), n ≥ 0} is a submartingale difference sequence, and a supermartingale iff {(U_n, F_n), n ≥ 0} is a supermartingale difference sequence.

(ii) A martingale difference sequence has constant expectation 0; a submartingale difference sequence has non-negative expectations; a supermartingale difference sequence has non-positive expectations.

Reference no: EM131830319

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