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For $\begin{Bmatrix} \zeta_t \end{Bmatrix}_{t\in [0,T]}$ a filtration on $(\Omega, \zeta,\mathbb{P})$ and $A^k=\begin{Bmatrix} A_t^k \end{Bmatrix}_{t\in [0,T]}$ a discrete and $\zeta_t$-adapted process, what is the reason to say that $A=(A_t^1,...,A_t^n)$ is adapted to filtration $\begin{Bmatrix} \zeta_t^{\bigotimes n} \end{Bmatrix}_{t\in [0,T]}$?

What is the meaning of $\bigotimes$? Simply a matrix product?

Thanks in advance for any clarification.

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The symbol $\otimes$ denotes the Kronecker product.

But, here the notation $\xi_t^{\otimes n}$ indicates the Kronecker power of $\xi_t$, which is defined as:

$$ \xi_t^{\otimes n} = \xi_t \otimes \xi_t^{\otimes (n-1)} = \xi_t \otimes \xi_t \otimes \xi_t \otimes \cdots \otimes \xi_t $$

e.g., for n = 4, we have

$$ \xi_t^{\otimes 4}= \xi_t \otimes \xi_t \otimes \xi_t \otimes \xi_t $$

we have also

$$ \xi_t^{\otimes 1}= \xi_t \qquad \text{and} \qquad \xi_t^{\otimes 0}= 1$$

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  • $\begingroup$ Incidentally, the Kronecker product is the usual tensor product in matrix form. $\endgroup$ Oct 14 '20 at 11:09

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