# Marginal Distribution of Diffusion Process

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!

• you can represent your process as an OU process and use its properties. en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process – thmusic Mar 10 at 17:04
• "[...]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 $a >0$, $b \in \mathbb{R}$)...? – saz Mar 10 at 17:33
• Does this mean it would just be a normal distribution with mean 2t and variance 3t? – Ryan B Mar 10 at 20:29
• The mean is correct but the variance is not. – saz Mar 11 at 8:18
• Oops, forgot to square. Think I have it now. Thank you! – Ryan B Mar 11 at 17:40