 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/m/1/b/iv/solution

Take $Y = U X_{1}$ , where

U

is uniform on

(0, 1)

and independent of

X_{1}

. Conditioning on

U

gives

E e^{t U X_{1}} = \int_{0}^{1} E (e^{t u X_{1}}) d u = \int_{0}^{1} M_{1} (u t) d u .

For

t

near zero this is finite because the moment-generating function of

X_{1}

is finite on a neighborhood of zero; Tonelli's theorem justifies the conditioning integral for the nonnegative exponential.

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