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 on $\mathbb R^d$. 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 $$ \sup_{t \in [0, 1]} \mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ] \le \frac{c(1 + \mathbb E [ |X_0|^{p+1} ])}{\varphi (R)}, \quad \forall R >0 \text{?} $$

Above, $\varphi: \mathbb R_+ \to \mathbb R_+$ is an increasing function and the constant $c>0$ possibly depends on $d, b, \sigma, p$. Thank you so much for your elaboration!

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