Since we can apply Burkholder-Davis-Gundy to control the supremum of moments, we start by removing the tail simply using $1\leq \frac{|X_{t}|}{R}$.

$$ \mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ]\leq \frac{1}{R}\mathbb E [ |X_t|^{p+1} ]. $$

And then as done in René Schilling, Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Chapter 19 (2nd edition) (see Application of the Burkholder Davis Gundy inequality) we have the bound

$$\sup_{t\leq T}\mathbb E [ |X_t|^{p+1} ]\leq \mathbb{E}\sup_{t\leq T}|X_t|^{p+1}\leq Ce^{CT}(1+\mathbb{E}|X_0|^{p+1}).$$

Since we can apply Burkholder-Davis-Gundy to control the supremum of moments, we start by removing the tail simply using $1\leq \frac{|X_{t}|}{R}$.

$$ \mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ]\leq \frac{1}{R}\mathbb E [ |X_t|^{p+1} ]. $$

And then as done in René Schilling, Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Chapter 19 (2nd edition) (see Application of the Burkholder Davis Gundy inequality) we have the bound

$$\sup_{t\leq T}\mathbb E [ |X_t|^{p+1} ]\leq \mathbb{E}\sup_{t\leq T}|X_t|^{p+1}\leq Ce^{CT}(1+\mathbb{E}|X_0|^{p+1}).$$

Interestingly, note that you can get faster decay by using $1\leq \frac{|X_{t}|^{q}}{R^{q}}$ if you have $\mathbb{E}|X_0|^{p+q}<\infty$.