I am working through some proofs related to differential equations and stochastic differential equations related to existence of solutions.
Consider a SDE
$$ dX_t = f(t,X_t)dt + dW_t $$
The following assumption I am unclear on
Assumption:
There exists $x_0 \in E$ such that $f(t,x_0) \in L^2[0,T]$ for all $0 < T <\infty$.
Why is this important for it to be integrable for the initial condition and not for all possible $x$?
I am assuming that f is Lipschitz also, which is the common condition for existence/uniqueness as in Shreve-Karatzas 5.2/2.9 theorem
Well because then you can obtain it for general x too
$$\int_{0}^{T}f^{2}(t,x)dt\leq \int_{0}^{T}c|x-x_{0}|^{2}dt+\int_{0}^{T}f^{2}(t,x_{0})dt<\infty,$$
where we used
$$|f(t,x)|^{2}=|f(t,x)-f(t,x_{0})+f(t,x_{0})|^{2}\leq |f(t,x)-f(t,x_{0})|^{2}+|f(t,x_{0})|^{2}.$$