I think I get stuck with the Heston SDE as following:
$X$ is a two dimensional adapted process satisfying the following SDE: $$ \alpha\ ,\beta\ >0\ , b(t,x)=(rx_1,\alpha(\beta-x_2))^T $$ $$\sigma(x)= \left(\begin{matrix} x_1\sqrt{x_2}&0 \\ \sigma\sqrt{x_2}\rho&\sigma\sqrt{x_2}\sqrt{1-\rho^2}\end{matrix}\right) $$
$$dX_t=b(t,X_t)·dt+\sigma(t,X_t)·dB_t,$$ where $2\alpha\beta\geq\sigma^2,\ t\in [0,T], \\ X_0=x\in D= (0,\infty)^2$.
Then how to prove the following estimates: $$E\left[\mathop{\sup}\limits_{t\in[0,T]}|X_t|^p \right]\leq C(1+|x|^p),\quad\forall p\geq 1.$$ The constant $C$ is related to $T$(maybe $p$?). I failed to prove it since the norm of $\sigma$ is not of linear growth! And I have already known that X does not go out of D since $2\alpha\beta\geq\sigma^2$, what I need is only the estimation!