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This question is influenced by this paper.

Consider a random field $Z(t,x)$ with a correlation structure given by

$$d\langle Z(t,x),Z(t,y)\rangle =c(x,y)\,dt.$$

For example, a choice for the correlation function can be $c(x,y)=ae^{-k|x-y|}.$

My question is, what is the meaning of the correlation structure $d\langle Z(t,x),Z(t,y)\rangle=c(x,y)\,dt$? Is $c(x,y)$ an instantaneous correlation between the random variables $Z(t,x)$ and $Z(t,y)$?. How (or why) is the cross-variation $\langle Z(t,x),Z(t,y)\rangle$ related to the correlation structure? Any help is appreciated!

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    $\begingroup$ On proper typesetting:$$ \begin{align} \text{wrong: }\qquad & d<Z(t,x),Z(t,y)>=c(x,y)dt \\ {} \\ \text{right: }\qquad & d\langle Z(t,x),Z(t,y)\rangle =c(x,y)\,dt \end{align} $$ (I've edited accordingly.) $\endgroup$ Jul 5, 2021 at 17:58
  • $\begingroup$ According to the papar,the identity is the very definition of the “correlation” $c$ . $\endgroup$ Jul 8, 2021 at 7:32
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    $\begingroup$ It means that $Z(t,x)Z(t,y)-c(x,y)t$ is a (local) martingale. $\endgroup$
    – Tobsn
    Jul 9, 2021 at 18:47
  • $\begingroup$ It's the sharp bracket/compensator: math.stackexchange.com/questions/902886/… $\endgroup$
    – jacques
    Jul 14, 2021 at 18:57

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