Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 4 / a

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- Solution a

Solution

Continuity gives

X_{τ_{a}} = a

on

{τ_{a} < \infty}

. The stopped process

X^{τ_{a}}

is bounded by

a

and is therefore a true martingale. Hence

1 = E X_{t \land τ_{a}} = a P (τ_{a} \leq t) + E [X_{t} 1_{{τ_{a} > t}}] .

The second term tends to zero by bounded convergence because

X_{t} \to 0

and it is bounded by

a

. Thus

P (τ_{a} < \infty) = P (t \geq 0 sup X_{t} > a) = \frac{1}{a} .

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