Decay estimate of moment of an SDE

Question

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!

Answer 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 $\mathbb{E}|X_0|^{p+q}<\infty$.