Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2008 / A / 2 / c

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- Solution c

Solution

Taking the logarithm of the explicit solution gives a linear function of the normal random variable

B_{t}

:

lo g X_{t} = lo g x + (r - σ^{2} /2) t + σ B_{t} .

Thus

X_{t} \sim Lognormal (lo g x + (r - σ^{2} /2) t, σ^{2} t) .

For

z > 0

its density is

\frac{1}{z σ 2 π t} exp [- \frac{( l o g ( z / x ) - ( r - σ ^{2} /2 ) t ) ^{2}}{2 σ ^{2} t}],

and it is zero for

z \leq 0

. The factor

1/ z

is the Jacobian of the logarithmic change of variable.

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