We have martingales $(M^1,...,M^d)$. What does it mean if the covariation $(\langle M^j,M^i\rangle)$ is strictly positive definite?
Does someone know a reference?
What does it mean if I integrate with respect to this positive covariation $\langle M^i,M^j\rangle$ ?
Right from the start, that the (quadratic)-covariation is positive definite has notting to do with the given martingales $(M^1,...,M^d)$.
The covariation is a symmetric bilinear form. And for symmetric bilinear form beeing positive definite is a well know definition. But for beeing definite you can't just consider the covariation on the set of semimartingales (the biggest set for which the covariation is defined, as far as I know), as all finite variation processes have zero quadratic variation and thus the zero-value is not unique. But if you consider the equivalence class $V$ in which all processes are identified, iff their difference has zero quadratic variation, then the zero-value is unique. This means for all $X\in V$ $\langle X,X\rangle\geq0$ and equality iff $X$ is the zero-value in $V$.
On this space $V$ the covariation is a positive definite symmetric bilinear from, which means it is a scalar product. Thus $V$ equiped with $\langle \cdot,\cdot\rangle$ is a inner product space.