1

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!

Michael Hardy
  • 1
Heisenberg
  • 3,387
  • 1
    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.) – Michael Hardy Jul 05 '21 at 17:58
  • According to the papar,the identity is the very definition of the “correlation” $c$ . – Ѕᴀᴀᴅ Jul 08 '21 at 07:32
  • 3
    It means that $Z(t,x)Z(t,y)-c(x,y)t$ is a (local) martingale. – Tobsn Jul 09 '21 at 18:47
  • It's the sharp bracket/compensator: https://math.stackexchange.com/questions/902886/angle-bracket-and-sharp-bracket-for-discontinuous-processes – jacques Jul 14 '21 at 18:57

0 Answers0