Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.

Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain Monrad condition holds if:

$$|\sigma^2(s,t)|\le \omega(s,t)^{1/\rho}$$

This is, loosely, equivalent to saying that the on diagonal $\rho$ variation is finite:

$$v_\rho(\sigma^2,I)=\sup_{(t_k)}\left(\sum |\sigma^2(t_k,t_{k+1})|^\rho \right)^{1/\rho}<\infty $$

See for example page 11 here: https://arxiv.org/pdf/1307.3460.pdf

I understand general Gaussian processes can be quite exotic so we must constrain them somehow. JM is needed to lift Gaussian processes to rough paths. But why is this the right condition? Why not some other condition? What is this function $\omega$ telling us? (If the on diagonal $\rho$ variation is finite, then it's simply that, but what is that variation telling us?)


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