 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2010/a/2/c/solution

f (x) = x^{4}

gives

d (B_{t}^{4}) = 4 B_{t}^{3} d B_{t} + 6 B_{t}^{2} d t .

The stochastic integral has expectation zero: its square integrability on a finite horizon follows from the finite sixth moments of a normal distribution. Since

E B_{s}^{2} = s

, integration and expectation yield

E B_{t}^{4} = 6 \int_{0}^{t} s d s = 3 t^{2} .

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