Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 1 / d

Solution

Use deterministic times tending to infinity and the Fatou lemma:

D := E [e^{M_{\infty} /2}] \leq n \to \infty lim inf E [e^{M_{n} /2}] \leq K < \infty.

X = a M

and

r = 1/ a

. Part (c) gives

E [E (a M)_{\infty}] = 1

, so

E [E (M)_{\infty}] \geq D^{- 2 (1/ a - 1)} .

Letting

a ↑ 1

shows that the terminal expectation is at least one. The nonnegative local martingale

E (M)

starts at one and is a supermartingale, so the Fatou lemma gives the opposite inequality. Consequently

E [E (M)_{\infty}] = 1.

E [E (M)_{\infty} ∣ F_{t}] \leq E (M)_{t} .

The left side has expectation one and the right side at most one, so equality holds almost surely. Thus

E (M)_{t} = E [E (M)_{\infty} ∣ F_{t}],

E (M)

is a uniformly integrable martingale. This establishes the required stopped-moment form of the Kazamaki criterion.

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