Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 35 / 3 / c

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- Solution c

Solution

Localize to compact subintervals of

(0, \infty)

, so that the Itô formula can be applied to

f (z) = z

. Its derivatives are

f^{'} (z) = 1/ (2 z)

and

f^{''} (z) = - 1/ (4 z^{3/2})

, while

d [Z]_{t} = 4 Z_{t} d t

. Thus, before

ζ

,

d R_{t} = \frac{1}{2 R _{t}} (2 R_{t} d B_{t} + γ d t) - \frac{1}{8 R _{t}^{3}} (4 R_{t}^{2} d t) = d B_{t} + \frac{γ - 1}{2 R _{t}} d t .

Hence the Bessel process equation is

d R_{t} = d B_{t} + \frac{γ - 1}{2 R _{t}} d t, R_{0} = r, t < ζ .

The inverse-power drift is interpreted only while

R > 0

; its boundary behaviour is justified next, not assumed in this calculation.

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