Consider the process X(t) = ae −αt + σ 2 B(t), t ≥ 0, where a, α ∈ R and σ > 0. Find the mean and covariance functions for this process. Show that X is a Gaussian process by applying Theorem 5. Does X...

Consider the process X(t) = ae^−αt
+ σ²B(t), t ≥ 0, where a, α ∈ R and σ > 0. Find the mean and covariance functions for this process. Show that X is a Gaussian process by applying Theorem 5. Does X have independent increments, and are these increments stationary? Is X a martingale, submartingale or supermartingale with respect to B?