Expectation of a product of Brownian Motions

[jamesa00789]

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.

 

[chiro]

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.

 

[jamesa00789]

Yes they are of the same standard brownian motion at different time intervals.

 

[chiro]

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?