This is a follow-up of a previous question I asked.
Suppose that $f \colon \mathbb R^n \to \mathbb R^n$ is a bounded vector field (not necessarily the gradient of another function), that $(W_t)_{0\leq t \leq T}$ is a standard Brownian motion in $\mathbb R^n$, and let $X$ denote the stochastic integral $$ X = \int_0^T f(W_t) \cdot d W_t. $$ Is it possible in general to bound conditional expectations of the following form in terms of $x \in \mathbb R^n$? $$ \mathbb E [X | W_T = x] $$ In particular, is the conditional expectation finite?