Consider a problem and its proof at the link. The problem boils down to what follows: prove that a pathwise-continuous Gaussian process with independent stationary increments has continuous mean and variance functions. Obviously, the pathwise-continuity condition can be replaced by the condition of continuity in probability. But is it true that the continuity condition can be completely dropped? Or there exists a Gaussian process with independent stationary increments that has discontinuous mean and variance?
Any help would be appreciated.
Best Answer
Let $X_t$ be a Gaussian process with independent stationary increments (Gaussian Levy process). Set \begin{align*} a(t) &:= \mathbb{E}[X_t]\\ b(t) &:= \mathrm{Var}[X_t] \end{align*} Note that $a,b$ are additive functions and $b$ is nondecreasing. Because the graph of an additive nonlinear function is dense in $\mathbb{R}^2$, $b$ must be linear, thus continuous. But $a$ can be discontinuous. In fact, the degenerate process $X'_t := \mu(t)$, where $\mu$ is any additive nonlinear function, is exactly an example of a Gaussian Levy process that has discontinuous mean.