Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 2 / d

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- Solution d

Solution

The Cameron-Martin theorem for a linear drift changes the density of Brownian paths through time

t

by

exp (b B_{t} - \frac{1}{2} b^{2} t) .

At the driftless hitting time

τ_{a, 0} = t

, the endpoint is

B_{t} = a

. Multiplying its given density by the likelihood

e^{ab - b^{2} t /2}

therefore yields

a (2 π t^{3})^{- 1/2} exp (- \frac{a ^{2}}{2 t} + ab - \frac{1}{2} b^{2} t) = a (2 π t^{3})^{- 1/2} exp (- \frac{( a - b t ) ^{2}}{2 t}) .

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