Expectation of a product of Brownian Motions

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Discussion Overview

The discussion revolves around the expectation of the product of standard Brownian motions, specifically E[Bt1.Bt2.Bt3], where Bt1, Bt2, and Bt3 are evaluated at different time intervals. The scope includes theoretical exploration and mathematical reasoning related to properties of Brownian motion.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant asks for the expectation E[Bt1.Bt2.Bt3] involving standard Brownian motions.
  • Another participant inquires about the independence of the Brownian motions and whether they refer to different intervals or have overlapping intervals.
  • A later reply clarifies that the Brownian motions are from the same standard Brownian motion but at different time intervals.
  • Another participant suggests using the definition of Brownian motion and the property E[XY] = E[X]E[Y] if the intervals are non-overlapping, and proposes decomposing overlapping intervals into non-overlapping parts.

Areas of Agreement / Disagreement

Participants generally agree that the Brownian motions are from the same process at different time intervals, but there is no consensus on how to handle overlapping intervals or the implications for calculating the expectation.

Contextual Notes

There are unresolved assumptions regarding the nature of the intervals (overlapping vs. non-overlapping) and how they affect the calculation of 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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