Quadratic variation of a scaled Itô integral

Let Xt=∫0ts dWsX_t = \int_0^t s\,dW_s, a stochastic integral against standard Brownian motion with a time-varying integrand. What is the quadratic variation [X]3[X]_3 of XX over [0,3][0,3]? (to 3 decimals)

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  1. For Xt=∫0tσs dWsX_t=\int_0^t \sigma_s\,dW_s, the quadratic variation is ∫0tσs2 ds\int_0^t \sigma_s^2\,ds.
  2. Square the integrand ss first, then integrate: ∫03s2 ds\int_0^3 s^2\,ds.

Answer

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9 (± 0.001)

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