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Working on a problem that I'm having some trouble starting. I have $X_t = 2t + 3B_t$ for $t \ge 0$ where $B_t$ is a Brownian Motion. I want to find the marginal distribution of $X_t$, as well as $E(X_s * X_t)$ for $s < t$. I'm pretty sure for the marginal distribution I need to apply Ito's formula, but I'm not exactly sure where to do this. Any help appreciated, thanks!

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  • $\begingroup$ you can represent your process as an OU process and use its properties. en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process $\endgroup$ – thmusic Mar 10 at 17:04
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    $\begingroup$ "[...]I'm pretty sure for the marginal distribution I need to apply Ito's formula" No, you don't. If $U \sim N(0,\sigma^2)$ then what is the distribution of $aU+b$ (for constants $a >0$, $b \in \mathbb{R}$)...? $\endgroup$ – saz Mar 10 at 17:33
  • $\begingroup$ Does this mean it would just be a normal distribution with mean 2t and variance 3t? $\endgroup$ – Ryan B Mar 10 at 20:29
  • $\begingroup$ The mean is correct but the variance is not. $\endgroup$ – saz Mar 11 at 8:18
  • $\begingroup$ Oops, forgot to square. Think I have it now. Thank you! $\endgroup$ – Ryan B Mar 11 at 17:40

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