# Gaussian process & Brownian motion

1. Oct 20, 2008

### shan

1. The problem statement, all variables and given/known data
Let $$\{ B(t) \}_{t \geq 0}$$ be a standard Brownian motion and $$U \sim U[0,1]$$ and $${Y(t)}_t\geq0$$ be defined by $$Y(t) = B(t) + I_{t=U}$$. 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:
$$Y(t_r) - Y(t_{r-1}) = B(t_r) + I_{t_r=U} - B(t_{r-1}) - I_{t_{r-1}=U} =B(t_r) - B(t_{r-1}) + I_{t_r=U} - I_{t_{r-1}=U}$$

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

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