Let's say, that $B_t$, $t\geq0$ is standard Brownian motion (Wiener process). Let's define process $$X_t=B_t+B_{t^2}\text{, }t\geq0$$
I need to find its variance, covariance, find out if it's Gaussian and its stationarity.
I started from variance:
1) $EX_t=E(B_t+B_{t^2})=EB_t+EB_{t^2}=0$ ?
2) $cov(X_s,X_t)=E(X_sX_t)=E(X_s[X_t-X_s+X_s])=E(X_sX_t-X_{s^2}+X_{s^2})=\ldots$
any ideas how to show those things? How I can interpret sum of two Brownian motions?
Thanks for suggestions
Best Answer
Hints:
You have correctly computed $E X_t$.
Recall that $\mathrm{Cov}(B_t,B_s) = t \wedge s$, and that $\mathrm{Cov}(\cdot,\cdot)$ is linear in both components.
Your answer for the variance and distributions will depend on whether $t<1$, $t=1$, or $t>1$.
Recall that Brownian motion has multivariate normal finite distributions. In particular $(B_t,B_{t^2})$ is bivariate normal (you compute the mean and covariance matrix (easy))
If $N$ is multivariate normal and $A$ is an affine transformation (like $(x,y) \to x + y$ cough cough)), then $AN$ is multivariate normal.
The sum of (jointly) stationary processes is stationary, can you show this?