A Proof Using The Optional Stopping Theorem
Assume (1), (5), (8), (10), (11) and (12). Using assumption (1), define the sequence of random variables
Assumption (11) implies that the conditional expectation of Xn given Fn–1 equals E almost surely for every n ∈ N, hence (Mn)n∈N0 is a martingale with respect to the filtration (Fn)n∈N0 by assumption (12). Assumptions (5), (8) and (10) make sure that we can apply the optional stopping theorem, hence MN = SN – TN is integrable and
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(13)
Due to assumption (8),
and due to assumption (5) this upper bound is integrable. Hence we can add the expectation of TN to both sides of Equation (13) and obtain by linearity
Remark: Note that this proof does not cover the above example with dependent terms.
Read more about this topic: Wald's Equation
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