 Solution

ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2025/3/a/ii/solution

M_{t} = e^{X_{t}}

. Since

d [X]_{t} = σ (X_{t})^{2} d t

and

b = - σ^{2} /2

, the two drift terms cancel:

d M_{t} = e^{X_{t}} d X_{t} + \frac{1}{2} e^{X_{t}} d [X]_{t} = \frac{e ^{X_{t}}}{1 + X _{t}^{2}} d B_{t} .

The integrand is continuous and hence locally square integrable after stopping at bounded levels of

X

. Thus

M

M_{0} = 1

M_{t} = 1 + \int_{0}^{t} \frac{e ^{X_{s}}}{1 + X _{s}^{2}} d B_{s}, [M]_{t} = ⟨ M ⟩_{t} = \int_{0}^{t} \frac{e ^{2 X_{s}}}{( 1 + X _{s}^{2} ) ^{2}} d s .

The cancellation is the reason

e^{x}

is the relevant scale function for this diffusion.

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