Second moment of a Gaussian-kernel stochastic integral

Let $W_t$ be a standard Brownian motion. Find the variance of $X = \displaystyle\int_0^2 \sqrt{t}\,e^{W_t^2/8}\,dW_t$ .

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Itô isometry: $\operatorname{Var}(X)=\mathbb{E}[\int_0^2 t\,e^{W_t^2/4}\,dt]=\int_0^2 t\,\mathbb{E}[e^{W_t^2/4}]\,dt$ .
$\mathbb{E}[e^{W_t^2/4}]=(1-t/2)^{-1/2}$ ; then $\int_0^2 t(1-t/2)^{-1/2}dt=\tfrac{16}{3}$ .

Answer

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5.3333 (± 0.02)

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