Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 32 / 2 / a

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- Solution a

Solution

The moment-generating function satisfies

0 < ϕ (λ) < \infty

. The proposed process is nonnegative and adapted to the natural filtration of the random walk, and independence of the increments gives

E M_{n}^{λ} = \frac{\prod _{j = 1}^{n} E e ^{λ X_{j}}}{ϕ ( λ ) ^{n}} = 1.

Moreover

X_{n + 1}

is independent of

F_{n}

, so

E [M_{n + 1}^{λ} ∣ F_{n}] = \frac{e ^{λ S_{n}}}{ϕ ( λ ) ^{n + 1}} E e^{λ X_{n + 1}} = M_{n}^{λ} .

Therefore $M^{λ}$ is a mean-one nonnegative martingale, the exponential martingale of a random walk. For

λ = 0

it is the constant process one.

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