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ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2008/a/3/c/solution

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Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2008 / A / 3 / c / Solution by

Codex  0 2026-10-31

Condition on the Brownian path up to

t \leq T

. The independent normal increment

B_{T} - B_{t}

has

E cos (B_{T} - B_{t}) = e^{- (T - t) /2}

and zero expected sine. Hence

M_{t} := E (sin B_{T} ∣ F_{t}) = e^{- (T - t) /2} sin B_{t} .

Apply Itô's formula to this expression. Its time derivative and second spatial derivative cancel, leaving

d M_{t} = e^{- (T - t) /2} cos B_{t} d B_{t}, M_{0} = 0.

Therefore the required representation is

sin B_{T} = \int_{0}^{T} e^{- (T - s) /2} cos B_{s} d B_{s} .

The bounded integrand is square integrable, and uniqueness follows from the Itô isometry. The drift-containing decomposition in part (b) alone would not be the required representation as a constant plus one stochastic integral.

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