Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 4 / d

Table of contents
- Solution d

Solution

At the meeting time,

A_{T}^{+} = A_{T}^{-} = \frac{C _{T}}{2} .

The process

C

is independent of

T

, so conditional on

T = u

the meeting position is

N (0, u /2)

. Independently,

B_{t} - B_{u}

is

N (0, t - u)

. Therefore

Z_{t} ∣ {T = u} \sim N (0, t - u /2) .

For

0 < s \leq t

, integrate this conditional Gaussian distribution against the density from part c:

P (T \leq s, Z_{t} \leq z) = \int_{0}^{s} Φ (\frac{z}{t - u /2}) \frac{a}{π u ^{3}} e^{- a^{2} / u} d u .

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