combination of brownian motion Suppose $B_t$ is a Brownian motion. As I understand, $B_2-B_1$ is independent of $B_3-B_2$ from properties of Brownian motion. Does it also mean that $B_1$ and $B_2$ are also independent? Can I use this independence to find the joint density of $B_1+B_2+B_3$ as each Brownian process is a normal process of mean 0 and variance t, it should be trivial.
I've another related question. To find the expectation over a Brownian process, can I integrate my stochastic process over the normal density function for Brownian motion (mean 0 and variance t)? I hope this makes sense.
 A: $B_1$ and $B_2$ are not independent. 
Since $B_1$ and $B_2-B_1$ are independent,
$$0=\mathrm{Cov}(B_1,B_2-B_1)=E[B_1 (B_2-B_1)]=E[B_1 B_2]-E[B_1^2]=E[B_1 B_2] - 1$$
So,
$$\mathrm{Cov}(B_1,B_2)=E[B_1 B_2]=1$$
You can use this result and the fact that linear combinations of normal variables are normal to calculate the distribution of $B_1+B_2+B_3$.
I'm not sure what exactly you meant by "expectation over a Brownian process". Can you please give us a little background of the problem you are trying to solve? Your method of taking an expectation with respect to $N(0,\sqrt{t})$ is fine when you want the expected value of some function of $B_t$. 
There is also a theory of stochastic integration with respect to stochastic processes which might be of use to you.
A: Expressing $B_1+B_2+B_3=3B_1+2(B_2-B_1)+(B_3-B_2)$ as a sum of independent normal random variables will help you find its density. I'm not sure why you call it a joint density.
A: I can not post comments, so I will post an answer..  Using $S_0 + B_t$ to model a stock price is not a very good idea, because $S_t$ may well turn negative. Have you considered using Geometric Brownian Motion ?
