References for estimates $E[|X_t|^{2n}]$ and $E[|X_t-X_{t_0}|^{2n}]$ of SDE

Asked Apr 14 '23 at 08:34

Active Apr 14 '23 at 12:27

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I'm reading this thread where the following results are mentioned.

Given a Ito-SDE $$dX_t=a(t,X_t)dt+b(t,X_t)dW_t$$ fulfilling the Lipshitz condition and linear growth condition. Then for some constants $C,D$:

$$ E\left[\sup_{t_0\le s \le T} |X_s|^{2n}\right]\le D \cdot (E[|X_{t_0}|^{2n}]+(1+E[|X_{t_0}|^{2n}]) \cdot (T-t_0)^n \exp(C(T-t_0))). $$
$$ E[|X_t|^{2n}] \le (1+E[|X_{t_0}|^{2n}]) \exp(C(t-t_0)). $$
$$ E[|X_t-X_{t_0}|^{2n}] \le D \cdot (1+E[|X_{t_0}|^{2n}]) \cdot (t-t_0)^n \cdot \exp(C(t-t_0)). $$

Could you elaborate on the references for the proofs of those estimates?

edited Apr 14 '23 at 12:27

asked Apr 14 '23 at 08:34

Akira

Yes, you can use Burkholder inequality and then Grownwall inequality eg. see here https://math.stackexchange.com/questions/1280256/application-of-the-burkholder-davis-gundy-inequality – Thomas Kojar Apr 17 '23 at 04:04

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