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Gaussian process & Brownian motion

  1. Oct 20, 2008 #1
    1. The problem statement, all variables and given/known data
    Let [tex]\{ B(t) \}_{t \geq 0}[/tex] be a standard Brownian motion and [tex]U \sim U[0,1][/tex] and [tex]{Y(t)}_t\geq0[/tex] be defined by [tex]Y(t) = B(t) + I_{t=U}[/tex]. Verify that Y(t) is a Gaussian process and state its mean and covariance functions. Is Y(t) a standard Brownian motion?

    2. Relevant equations

    3. The attempt at a solution
    The way I thought about doing this is look at the increments of Y(t), show that they are normally distributed, show that a random vector of increments of Y(t) is multivariate normal, conclude that Y(t) is Gaussian. If the mean is 0 and variance is t, then Y(t) is standard Brownian.

    So for the increment between time r and r-1:
    [tex]Y(t_r) - Y(t_{r-1}) = B(t_r) + I_{t_r=U} - B(t_{r-1}) - I_{t_{r-1}=U}
    =B(t_r) - B(t_{r-1}) + I_{t_r=U} - I_{t_{r-1}=U}[/tex]

    Where the first two terms are Normal(0,1) and the last two terms have this distribution:
    [tex]0 P(t_r \ne U \cap t_{r-1} \ne U)
    -1 P(t_r \ne U \cap t_{r-1} = U)
    1 P(t_r = U \cap t_{r-1} \ne U)[/tex]

    I know that the above is still normally distributed but I don't know how to work out the mean and variance. Am I on the right track?
    Last edited: Oct 20, 2008
  2. jcsd
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