Given a general SDE: $dX_t=b(X_t,t)dt+\sigma (X_t,t)dB_t$ , $X_0 =x$ and a solution $X^x_t$, where $b|x-y|+\sigma |x-y|\leq |x-y|$ .
Prove that: $\mathbb E(X^x_t)^2\leq L \exp(Lt)$ for some none negative $L$ and for all $t > 0$.
If $X^n_t$ are by the Picard Iterations(Cause I have no other way of describing a solution), then their limit is a solution and each of them is measurable by the filtration. I want to apply Doob's optional stopping time theorem on $\mathbb E(X^x_t)^2$ . For this I need a supermartingale and at least one of the boundary conditions in the theorem.
Editing further:
Now I have:
$\mathbb E\,[\sup\limits_{t\leq s}|X_{n}(s)-X_{n-1}(s)|^2] \leq$
$\leq T^2\mathbb E[|X_n(s)-X_{n-1}(s)|^2]+K\,T\,\mathbb E\,[|X_n(s)-X_{n-1}(s)|^2]$
And I think that my next step is applying some sort of inequality (Doob's/Jensen/etc..) But I'm not sure which one and how, any help?
Hint: start off by looking at $\mathbb{E}[\sup_{s\leq t}|X_{s}|^{2}]$. You should be able to bound it above by a sum of three expectations involving the starting point, the contributions from the drift, and the Ito integral. In particular, the Ito integral part should look like $$\mathbb{E}\left[\sup_{s\leq t}\left|\int_{0}^{s}\sigma(X_{s},s)dW_{s}\right|^{2}\right].$$