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Proving a process is Brownian Motion

  1. Dec 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Is the process [itex]\{X(t)\}_{t\geq 0}[/itex], where [itex]X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t)[/itex] Standard Brownian Motion?

    Where [itex]\rho\in(0,1), \ B_1(t)[/itex] and [itex]B_2(t) [/itex] are independent standard brownian motions


    2. Relevant equations



    3. The attempt at a solution

    Obviously [itex]X(0)=0[/itex]. Now let [itex]0\leq t_1<t_2<t_3[/itex]. Then, I can show that [itex]X(t_2)-X(t_1)\sim\mathcal{N}(0, t_2-t_1)[/itex]. My problem is showing that [itex]X(t_3)-X(t_2)[/itex] and [itex]X(t_2)-X(t_1)[/itex] are independent. I can show their covariance is 0, but that only implies independence if the process is Gaussian, which I have not shown. Any help would be much appreciated!
     
  2. jcsd
  3. Dec 16, 2012 #2

    Ray Vickson

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    For i = 1,2 and j = 1,2, are the ##B_i(t_3)-B_i(t_2)## independent of the ##B_j(t_2)-B_j(t_1)##?
     
  4. Dec 16, 2012 #3
    Hi Ray, yes they are.
     
  5. Dec 16, 2012 #4
    Not sure how to proceed.
     
  6. Dec 16, 2012 #5
    Are you implying that if [itex]A\perp B\perp C \perp D[/itex], then [itex]A+B \perp C+D[/itex], where [itex]\perp[/itex] means independent?
     
  7. Dec 17, 2012 #6

    Ray Vickson

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    More than that: if [itex]A\perp B\perp C \perp D[/itex], then [itex] f(A,B) \perp g(C,D)[/itex] for any (measurable) functions f(.,.) and g(.,.).
     
  8. Dec 17, 2012 #7
    Thanks. How can I go about proving that?
     
  9. Dec 17, 2012 #8

    Ray Vickson

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    It depends on what you know already. Anyway, all you need is the result for A+B and C+D, and you can use any number of standard tools, but first you need to know what those tools are. Do you know about characteristic functions? What can you say about the density (or cdf) of a set of independent random variables? How can you test whether two random variables (A+B and C+D in this case) are independent?

    All of these things can be found in books and articles---many freely available on-line---so I will leave it to you to search for the answers if you do not know them already. If you do know them already, just think about how you would use them.
     
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