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Consider $$ G_t = \int_0 ^t \frac{B_u}{u}du$$ where $\left(B_{t} \right)_{t\geq0}$ is $\mathcal F _t $ - brownian motian in $\mathbb R$, null at the origin. It's simple to show that $\left(G_{t} \right)_{t\geq0}$ is a centred gaussien process. But, when it comes to the covariance evaluation, it seems we can find some probles with singularity at zero. Someone could help me ? |
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For every nonnegative $t$, $$\mathbb E(G_t^2)=2\int_0^t\int_0^s\mathbb E(B_uB_s)\frac{\mathrm du}u\frac{\mathrm ds}s=2\int_0^t\int_0^su\frac{\mathrm du}u\frac{\mathrm ds}s=2\int_0^ts\frac{\mathrm ds}s=2t. $$ Likewise, for every nonnegative $s\leqslant t$, $$\mathbb E(G_s(G_t-G_s))=\int_s^t\int_0^s\mathbb E(B_uB_v)\frac{\mathrm du}u\frac{\mathrm dv}v=\int_s^t\int_0^su\frac{\mathrm du}u\frac{\mathrm dv}v=s\log\left(\frac{t}s\right). $$ Thus, for every nonnegative $s\leqslant t$, $$ \mathrm{Cov}(G_s,G_t)=2s+s\log\left(\frac{t}{s}\right). $$ |
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