 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2010/a/1/b/solution

Use

s

for the dummy integration variable in the printed integral. The Itô isometry gives

E ∣ z_{t}^{(n + 1)} ∣^{2} = C \int_{0}^{t} E ∣ z_{s}^{(n)} ∣^{2} d s \leq \frac{C ^{n + 1}}{n !} \int_{0}^{t} s^{n} d s = \frac{C ^{n + 1} t ^{n + 1}}{( n + 1 )!} .

The assumed second-moment bound also proves the integrability needed to apply the Itô isometry. The extra time integration is the source of the factorial improvement.

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