 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2007/a/1/c/solution

B_{t} - B_{s}

is independent of

F_{s}

e^{- ξ^{2} (t - s) /2}

. Since

B_{s}

s

E (e^{i ξ B_{t}} ∣ F_{s}) = e^{i ξ B_{s}} e^{- ξ^{2} (t - s) /2} .

The heat kernel identity in the hint gives

(P_{u} f) (x) = e^{i ξ x} e^{- ξ^{2} u /2}

, so this is exactly

(P_{t - s} f) (B_{s})

. Multiplying by the deterministic factor

e^{ξ^{2} t /2}

gives

E (M_{t} ∣ F_{s}) = M_{s} .

The process is adapted, and

∣ M_{t} ∣ = e^{ξ^{2} t /2}

is finite at each fixed time. Thus it is a complex martingale, meaning that its real and imaginary parts are integrable martingales.

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