 Solution

ID: past-exam-of-the-continuous-martingales-and-stochastic-calculus-course-of-the-university-of-oxford/2023/2/c/iii/solution

Part (ii) represents the process as an Iterated Itô integral:

B_{t}^{3} - 3 t B_{t} = 6 I_{3} (t) = 3 \int_{0}^{t} (B_{s}^{2} - s) d B_{s} .

B_{s}

s

E (B_{s}^{2} - s)^{2} = 3 s^{2} - 2 s^{2} + s^{2} = 2 s^{2} .

Consequently, for every finite

T

E \int_{0}^{T} (B_{s}^{2} - s)^{2} d s = \frac{2 T ^{3}}{3} < \infty.

[0, T]

. Since

T

is arbitrary,

B_{t}^{3} - 3 t B_{t} is a martingale, E (B_{t}^{3} - 3 t B_{t})^{2} = 6 t^{3} .

The last identity is the Itô isometry.

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