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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/1/b/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 / 1 / b / Solution by

Codex  0 2026-10-31

Use a filtration with respect to which

B

is Brownian. The process is adapted and continuous because it is a continuous function of

B_{t}

and

t

. The moment-generating function gives

E M_{t} = 1, E M_{t}^{2} = e^{- α^{2} t} E e^{2 α B_{t}} = e^{α^{2} t} < \infty.

For

0 \leq s < t

, write

M_{t} = M_{s} exp {α (B_{t} - B_{s}) - α^{2} (t - s) /2} .

The increment is independent of

F_{s}

and normal with variance

t - s

, so the conditional expectation of the second factor is one. Thus

E (M_{t} ∣ F_{s}) = M_{s} .

This verifies the martingale property from the definition and proves square integrability at each finite time, as well as continuity. It does not assert uniform square integrability over an infinite time horizon.

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