 Solution

ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2025/2/a/i/solution

For

t > 0

B_{t}

N (0, t)

L (t, α, 0) = \frac{1}{2 π t} \int_{R} exp [- (α + \frac{1}{2 t}) x^{2}] d x = \frac{1}{1 + 2 α t} .

t = 0

the defining random variable is one, consistent with the same formula. Thus

L (t, α, 0) = (1 + 2 α t)^{- 1/2} (t, α \geq 0) .

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