Variance of an Itô integral

HardStochastic Processes~8m

Let $W_t$ be a standard Brownian motion. The variance $\operatorname{Var}\!\left(\int_0^t W_s\,dW_s\right)$ has the form $k\,t^2$ . Find the constant $k$ .

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$\int_0^t W_s\,dW_s = \tfrac12(W_t^2 - t)$ ; the constant doesn't affect variance.
Use $\operatorname{Var}(W_t^2)=\mathbb{E}[W_t^4]-t^2=3t^2-t^2=2t^2$ . (Or apply Itô isometry.)

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Asked at: Jane Street, Two Sigma

Variance of an Itô integral

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