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?