Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 2 / c

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- Solution c

Solution

Condition on

S

and use the characteristic function of a standard normal distribution:

E_{Z} [S^{i Z} ∣ S] = E_{Z} [e^{i Z l o g S}] = e^{- (l o g S)^{2} /2} = G (S) .

The integrand has modulus one, so Fubini's theorem is immediate. Taking expectation over

S

yields

E [G (S)] = E [M (i Z)] .

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