Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 33 / 4 / b

Table of contents
- Solution b

Solution

The moment-generating function of a normal distribution gives

E e^{λ X_{1}} = exp (- λ + \frac{λ ^{2}}{2}) .

Independence of

X_{n + 1}

from

F_{n}

implies

E [e^{λ S_{n + 1}} ∣ F_{n}] = e^{λ S_{n}} exp (- λ + \frac{λ ^{2}}{2}) .

Every exponential here is integrable. Thus the process is a martingale exactly when

- λ + λ^{2} /2 = 0

. Its two roots are

0

and

2

, and the unique positive root is

λ = 2.

The resulting exponential martingale of a random walk is nonnegative and has expectation

1

at every finite time.

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