Let $A_t$ be a stochastic process of symmetric positive definite $n \times n$ matrices and let $B_t$ be the standard $n$-dimensional Brownian motion.
Define the martingale $X_t =\int\limits_0^tA_sdB_s$. Ito's isometry tells us $$E[X_t\cdot X_t^T] = E\left[\left(\int\limits_0^tA_sdB_s\right)\left(\int\limits_0^tA_sdB_s\right)^T\right]= \int\limits_0^t E[A_s^2]ds.$$ (Here $X_t\cdot X_t^T$ is an $n \times n$ matrix).
I'm interested in 'higher powers' of the right hand side. Namely, what can be said about the following quantity? $$E\left[\left(\int\limits_0^t A_s^2ds\right)^2\right].$$ In the case $n = 1$, the Burkholder Davis Gundy inequality implies $$E\left[\left(\int\limits_0^t A_s^2ds\right)^2\right] \leq CE\left[\left(\int\limits_0^tA_sdB_s\right)^4\right] = CE[X_t^4], $$ For some constant $C >0$. What is the higher dimensional analogue (if any) of this inequality, references will be most welcome.