# Exercise Solution 4.22

To calculate *E*(* ^{t}X*), we start with [4.55]:

[4.55]

We assume the unconditional expectation exists, in which case *E*(^{t}^{–k}* X*) =

*E*(

^{t}

**) for all**

*X**k*. Taking the unconditional expectation of both sides of [4.55], we obtain

[s1]

[s2]

[s3]

[s4]

We have obtained an expression for *E*(* ^{t}X*) in terms of itself. Rearranging, we conclude

[s5]

To calculate ^{t–1}*E*(* ^{t}X*), we again start with [4.55]:

[4.55]

Taking the conditional expectation of both sides, we obtain

[s6]

[s7]

[s8]

[s9]