 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2008/a/2/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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