Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 2 / a / i

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- Solution i

Solution

For

t > 0

,

B_{t}

has the normal distribution

N (0, t)

. Evaluate its Gaussian integral:

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} .

At

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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