# Meaning of $\bigotimes$

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.

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

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

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