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Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process.

I think $\mathsf{E}[O_t]=e^{-\alpha t} \mathsf{E}\left[ \int^t_0 e^{\alpha s} dB_s\right] = e^{-\alpha t} \times 0 = 0$ as the Ito integral is a martingale with expectation $0$ by definition. But I am not sure why it is a martingale in the first place...
No idea how to show it's a Gaussian process.

Please help!

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    $\begingroup$ What would be suitable indications really depends on what you know about Brownian motion and Gaussian processes--a point about which you say nothing in your post. $\endgroup$
    – Did
    Oct 3, 2012 at 5:35
  • $\begingroup$ In fact, this Ito integral is for deterministic integrand, hence it coincides with ... .... and by the properties of jointly Gaussian r.v. one gets that $O_t$ is ... ... $\endgroup$
    – Tarasenya
    Oct 3, 2012 at 7:30

1 Answer 1

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The integrand in the ito integral is deterministic since exp(alpha*s) is not random. Therefore the ito integral is merely a linear combination of Brownian increments. Brownian increments are distributed gaussian mean 0 and variance equal to the length of the time interval. Therefore we have a linear combination of gaussian random variables which is gaussian.

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