1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?

Similar Discussions: Gaussian process & Brownian motion
  1. Stochastic Processes (Replies: 0)

  2. Gaussian Curvature (Replies: 0)