Integral: Is there a closed form?

Question

I wonder whether there is a closed form or way to compute explicitly: $$\int_0^t e^{\alpha s} dB_s$$ where $\alpha$ is just a real number and the integral is in the Itô sense. Thank you very much!

Answer 1

This is already a closed form... Of course this integral is nothing but a centered normal random variable with variance $\sigma^2_t=\int\limits_0^t\mathrm e^{2\alpha s}\mathrm ds$ and as such, it is equal in distribution to $\sigma_tB_1$ or to $B_{\sigma^2_t}$ or to a host of other gaussian random variables.