 Solution

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

The relation between

B

and

W

d B_{s} = d W_{s} - β B_{s} d s

with

B_{0} = 0

e^{β s}

d (e^{β s} B_{s}) = e^{β s} d W_{s} .

Hence

B_{s} = \int_{0}^{s} e^{- β (s - u)} d W_{u} = W_{s} - β \int_{0}^{s} e^{- β (s - u)} W_{u} d u .

B

as an explicit function of the continuous path of

W

, using only an ordinary time integral. This inversion holds before changing measure;

W

 New to topics? Read the docs here!