# Decay estimate of moment of an SDE

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!

• The bound should depend, non-multiplicatively, on the Lipschitz constants. Mar 3 at 15:38
• If by 'increasing' you mean 'increasing to infinity' then it's not even true for $b=\sigma=0$... Mar 3 at 15:43
• @IosifPinelis Could you please elaborate more? Mar 3 at 16:43
• @MartinHairer the exponent on the RHS should be $p+1$ rather than $p$. I have edited my thread to fix this typo... Mar 3 at 16:45
• Take e.g. $X_0=0$, $b=1$, $\sigma=0$. Mar 3 at 17:51

Since we can apply Burkholder-Davis-Gundy to control the supremum of moments, we start by removing the tail simply using $$1\leq \frac{|X_{t}|}{R}$$.
$$\mathbb E [ |X_t|^p 1_{\{|X_t| \ge R\}} ]\leq \frac{1}{R}\mathbb E [ |X_t|^{p+1} ].$$
$$\sup_{t\leq T}\mathbb E [ |X_t|^{p+1} ]\leq \mathbb{E}\sup_{t\leq T}|X_t|^{p+1}\leq Ce^{CT}(1+\mathbb{E}|X_0|^{p+1}).$$
Interestingly, note that you can get faster decay by using $$1\leq \frac{|X_{t}|^{q}}{R^{q}}$$ if you have $$\mathbb{E}|X_0|^{p+q}<\infty$$.