Let $(X_t)_t$ a stochastic process. We denote it's quadratic variation by $$\left<X\right>_t=\lim_{\|P\|\to 0}\sum_{i=1}^n(X_{t_{k+1}}-X_{t_k})^2,$$ where $P$ range over all partition of $[0,t]$.
The notation $\left<X_t\right>$ is commonly used for $\mathbb E[X_t]$ in physic. My question : Is there a link between $\left<X\right>_t$ and $\left<X_t\right>$ (i.e. the expectation of $t$) ?.
Same, we denote the quadric covariance of $(X_t)_t$ and $(Y_t)_t$ by $$\left<X,Y\right>_t=\lim_{\|P\|\to 0}\sum_{i=1}^n (X_{t_{k+1}}-X_t)(Y_{t_{k+1}}-Y_t).$$
We also denote $\left<X_t,Y_t\right>$ the covariance of $X_t$ and $Y_t$. Is there a link between $\left<X,Y\right>_t$ and $\left<X_t,Y_t\right>$ or al these are just notation, and there are absolutely no link ?