Edit of progress: Since the SDE is linear, I got a solution in the form of $e^\int...$$\cdot e^\int$. By Jensen's inequality I can change the order of the left factor to have the integral on the outside. Am I correct in my method, and if so what should I do next?
Let $X_t^x$ be a solution to the SDE: $dX_t=b(X_t)+\sigma(X_t)dW_t$ with $X_0=x$.
Assume that $b$ and $\sigma$ are Lipschitz Continuous.
I want to proof that there exists $0<L$ such that $E\left|X_t^x\right|^2$ $\leq$ $Le^L$$^t$.
I think it's called a uniform boundary.
I have tried to apply Gronwall's Lemma (from ODE) in several ways, but I'm not sure it applies to SDE.