Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 5 / i

Solution

Let

A = ⟨ M ⟩

be the quadratic variation and stop at

T_{b} = in f {u \geq 0 : A_{u} > b}

. Its continuity and monotonicity imply

A_{u \land T_{b}} \leq b

and

A_{T_{b}} = b

whenever

T_{b} < \infty

. The standard martingale isometry after stopping by bounded quadratic variation gives

E (M_{t \land T_{b}}^{2}) = E A_{t \land T_{b}} \leq b

; in particular the stopped local martingale is square-integrable. This criterion follows by localization and the Doob L2 maximal inequality, which bound the second moment of the stopped supremum by

4 b

and then permit removal of localization.

{A_{t} \leq b}

one has

T_{b} \geq t

, so the original and stopped paths agree through time

t

P (s \leq t sup M_{s} > a) \leq P (s \leq t sup ∣ M_{s \land T_{b}} ∣ > a) + P (A_{t} > b) \leq \frac{4 b}{a ^{2}} + P (A_{t} > b) .

Thus

P (s \leq t sup M_{s} > a) \leq 4 b / a^{2} + P (⟨ M ⟩_{t} > b) .

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