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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.

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up vote 4 down vote accepted

$B_1$ and $B_2$ are not independent.

Since $B_1$ and $B_2-B_1$ are independent, $$0=\mathrm{Cov}(B_1,B_2-B_1)=E[B_1 (B_2-B_1)]=E[B_1 B_2]-E[B_1^2]=E[B_1 B_2] - t_1 \sigma^2$$ So, $$\mathrm{Cov}(B_1,B_2)=E[B_1 B_2]=t_1 \sigma^2$$

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 $N(0,\sigma\sqrt{t})$ 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.

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Thanks. I have a stock price function that is a stochastic process (e.g. $S = S_0 + B_t$). Now I am interested in finding various option values over those stock prices which involves finding the expectation. So to find asian call value, I need to find $E(\frac{S_1+S_2}{2} - K)^+$ which requires finding the density function $B_1+B_2$ computed by differentiating the distribution function. Hope this makes sense. – user957 Oct 25 '10 at 1:08
You do need stochastic integration. There are many textbooks on applying stochastic calculus to option pricing. Look at Hull's Options (intermediate) or Duffie's Dynamic Asset Pricing Theory (advanced). Oksendal's Stochastic Differential Equations gives the math background with a chapter on finance applications. – Jyotirmoy Bhattacharya Oct 25 '10 at 1:13
@user957 curious how stock price is not mentioned in the original post. Maybe this is not the proper forum. – Ross Millikan Oct 25 '10 at 4:32

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.

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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 ?

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