Exercise Solution 4.22

To calculate E(^tX), we start with [4.55]:

[4.55]

We assume the unconditional expectation exists, in which case E(^t–kX) = E(^tX) for all k. Taking the unconditional expectation of both sides of [4.55], we obtain

[s1]

[s2]

[s3]

[s4]

We have obtained an expression for E(^tX) in terms of itself. Rearranging, we conclude

[s5]

To calculate ^t–1E(^tX), we again start with [4.55]:

[4.55]

Taking the conditional expectation of both sides, we obtain

[s6]

[s7]

[s8]

[s9]