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

1 Answer 1

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

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