We consider an SDE
$$
d X_t = b(t, X_t) dt + \sigma(t, X_t) d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion. We fix $p \in [1, \infty)$. Here $b, \sigma$ are Lipschitz in space uniformly in time.

>Are there decay estimates of moment in a form
$$
\sup_{t \in [0, 1]} \mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ] \lesssim \frac{\mathbb E [ |X_0|^{p+1} ]}{\varphi (R)},
\quad \forall R >0
$$
?

Above : $\varphi: \mathbb R_+ \to \mathbb R_+$ is an increasing function. Thank you so much for your elaboration!