 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/m/m4/e/ii/solution

For each fixed

θ

, the Taylor expansion of the exponential yields

lo g M_{U_{t}} (θ) = \frac{θ ^{2}}{2} + O ((λ t)^{- 1/2}) ⟶ \frac{θ ^{2}}{2} .

The limiting moment-generating function

e^{θ^{2} /2}

is that of the standard normal distribution and is finite around zero. The moment-generating function continuity theorem therefore gives

U_{t} d N (0, 1) .

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