Expectation of a product of Brownian Motions

AI Thread Summary
The discussion revolves around calculating the expectation E[Bt1.Bt2.Bt3] for standard Brownian motions Bt1, Bt2, and Bt3. Participants clarify that if the Brownian motions are independent, the expectation can be simplified using the property E[XY] = E[X]E[Y]. It is noted that since the motions refer to the same process at different time intervals, their independence is crucial for the calculation. If the intervals overlap, the approach involves decomposing the processes into non-overlapping parts. Ultimately, understanding the relationship between the intervals and the properties of Brownian motion is essential for solving the expectation.
jamesa00789
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Let Bt1, Bt2 and Bt3 be standard Brownian motions with ~N(0,1).

Then what is E[Bt1.Bt2.Bt3] ?

Any help would be much appreciated.
 
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jamesa00789 said:
Let Bt1, Bt2 and Bt3 be standard Brownian motions with ~N(0,1).

Then what is E[Bt1.Bt2.Bt3] ?

Any help would be much appreciated.

Hey jamesa00789 and welcome to the forums.

What are the conditions for each BM? Are they independent? Do they refer to different intervals for the same process? Maybe some overlap in intervals?

If they are truly independent you can use the property that E[XY] = E[X]E[Y] and take it from there.
 
Yes they are of the same standard brownian motion at different time intervals.
 
jamesa00789 said:
Yes they are of the same standard brownian motion at different time intervals.

If they are are at non-overlapping intervals, then use the definition of the Brownian motion. If they are over-lapping, then decompose it into processes that are non-overlapping and take care of parts that are overlapping.

Using this, the fact that E[XY] = E[X]E[Y], and the definition of BM, what do you get?
 

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