Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm : for all $t>0$, $\vert\vert A_t \vert\vert_{op} \leq b$ for some constant $b$. That is $$0\preceq A \preceq bI_n $$ Define, for all time $t\geq0$ the martingale $M_t=\int_0^sA_sdW_s$.

Is it true that both processes can be simultaneously confined ? That is for all $T>0$ and all $R>0$, $$\mathbb{P}(\forall t \in [0,T], W_t \in B(0,R) \text{ and }\forall t \in [0,T], M_t \in B(0,R)) > 0 \quad? $$ For my purpose I would actually need the slightly weaker :

$$\mathbb{P}(W_T \in B(0,R) \text{ and }\forall t \in [0,T], M_t \in B(0,R)) > 0 $$ Fix some $T>0$. I tried to work on the event $E_R = \{\forall t \in [0,T], W_t \in B(0,R)\}$ but it is unclear to me how $M_t$ conditionned to that event behaves, as there could be cancellations assuring that $W_t$ is small that might be destroyed in the stochastic integral right ? On the other hand it looks plausible to me that $\omega \in E_R \implies M_t(\omega) \in B(0,\lambda R) $ for some $\lambda$ potentially large but I might be wrong ?

EDIT : The general answer is no, as shown in an answer below. What kind of hypothesis on $A_t$ could fix the issue ? If $A_t$ is an homotethy for instance it is of course true. Is :

$$aI_n\preceq A \preceq bI_n \quad \text{ almost surely}$$ for some $a,b>0$ enough for instance ?