I need to find the autocovariance $C_{YY}(t,s)$ of the stochastic process $Y(t) = t^2 X(t) -2X'(t)$ where $C_{XX}(t,s) = e^{-t^2 -s^2}$ is given.
Using known properties I can calculate the autocovariance of both $Y_1(t)=t^2 X(t)$ and $Y_2(t)=-2 X'(t)$, but how about the sum of these?
To restate the problem: it is noticeable (by definition of autocovariance) that $C_{YY}(t,s) \neq C_{Y_1 Y_1}(t,s) + C_{Y_2 Y_2}(t,s)$, but what is missing?
By assuming regularity conditions that allow interchange between limit and expectation you can calculate the crosscovariance of X and its derivative: $C_{X,X'}(t,s)=E[(X(t)-E[X(t)])(\lim_{h\rightarrow0}{X(s+h)-E[X(s+h)] -(X(s)-E[X(s)])})=\lim_{h\rightarrow0}\frac{C_{X,X'}(t,s+h)-C_{X,X'}(t,s)}{h}=\frac{\partial C_{X,X}(t,s)}{\partial s}$ The resulting autocovariance is: $C_{Y,Y}(t,s)=C_{Y_1,Y_1}(t,s)+C_{Y_2,Y_2}(t,s)+C_{Y_1,Y_2}(t,s)+C_{Y_1,Y_2}(s,t)$ where, $C_{Y_1,Y_1}(t,s)=(st)^2C_{X,X}(t,s) \\ C_{Y_2,Y_2}(t,s)=4C_{X,X}\frac{\partial^2 C_{X,X}(t,s)}{\partial t\partial s} \\ C_{Y_1,Y_2}(t,s)=-2t^2\frac{\partial C_{X,X}(t,s)}{\partial s}$